The Quantification of Beliefs, from Bayes to A.I., and Its Consequence on the Scientific Method

The Quantiﬁcation of Beliefs, From Bayes to A.I., And its Consequence on the Scientiﬁc Method

Frédéric Galliano

AIM, CEA/Saclay, France 2 BAYES’ RULE THROUGH HISTORY Early Development The Frequentist Winter The Bayesian Renaissance

3 IMPLICATIONS FOR THE SCIENTIFIC METHOD Karl Popper’s Logic of Scientiﬁc Discovery Bayesian Epistemology How Researchers Actually Work

4 SUMMARY & CONCLUSION

Outline of the Talk

1 BAYESIANS VS FREQUENTISTS Epistemological Principles & Comparison Demonstration on a Simple Example Limitations of the Frequentist Approach

F. Galliano (AIM) Astromind 2019, CEA/Saclay 2 / 36 3 IMPLICATIONS FOR THE SCIENTIFIC METHOD Karl Popper’s Logic of Scientiﬁc Discovery Bayesian Epistemology How Researchers Actually Work

4 SUMMARY & CONCLUSION

Outline of the Talk

1 BAYESIANS VS FREQUENTISTS Epistemological Principles & Comparison Demonstration on a Simple Example Limitations of the Frequentist Approach

2 BAYES’ RULE THROUGH HISTORY Early Development The Frequentist Winter The Bayesian Renaissance

F. Galliano (AIM) Astromind 2019, CEA/Saclay 2 / 36 4 SUMMARY & CONCLUSION

Outline of the Talk

1 BAYESIANS VS FREQUENTISTS Epistemological Principles & Comparison Demonstration on a Simple Example Limitations of the Frequentist Approach

2 BAYES’ RULE THROUGH HISTORY Early Development The Frequentist Winter The Bayesian Renaissance

3 IMPLICATIONS FOR THE SCIENTIFIC METHOD Karl Popper’s Logic of Scientiﬁc Discovery Bayesian Epistemology How Researchers Actually Work

F. Galliano (AIM) Astromind 2019, CEA/Saclay 2 / 36 Outline of the Talk

1 BAYESIANS VS FREQUENTISTS Epistemological Principles & Comparison Demonstration on a Simple Example Limitations of the Frequentist Approach

2 BAYES’ RULE THROUGH HISTORY Early Development The Frequentist Winter The Bayesian Renaissance

3 IMPLICATIONS FOR THE SCIENTIFIC METHOD Karl Popper’s Logic of Scientiﬁc Discovery Bayesian Epistemology How Researchers Actually Work

4 SUMMARY & CONCLUSION

F. Galliano (AIM) Astromind 2019, CEA/Saclay 2 / 36 Outline of the Talk

1 BAYESIANS VS FREQUENTISTS Epistemological Principles & Comparison Demonstration on a Simple Example Limitations of the Frequentist Approach

2 BAYES’ RULE THROUGH HISTORY Early Development The Frequentist Winter The Bayesian Renaissance

3 IMPLICATIONS FOR THE SCIENTIFIC METHOD Karl Popper’s Logic of Scientiﬁc Discovery Bayesian Epistemology How Researchers Actually Work

4 SUMMARY & CONCLUSION

F. Galliano (AIM) Astromind 2019, CEA/Saclay 3 / 36 p(A and B) = . p(B)

⇔ probability of A, within B (new Universe): (2) p(A|B) = (2) + (3)

= p(B|A)p(A)

p(B|A)p(A) ⇒ p(A|B) = . p(B)

p(SNIa|binary) ' 10−12 yr−1 while p(binary|SNIa) = 1.

Conditional probability: p(A|B): probability of A, knowing B

Bayes’ rule (general probability theorem): By symmetry of A and B: p(A and B) = p(A|B)p(B)

Example of p(A|B) 6= p(B|A):

Prologue: Visual Representation of Bayes’ Rule (Venn Diagram)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 4 / 36 p(A and B) = . p(B)

⇔ probability of A, within B (new Universe): (2) p(A|B) = (2) + (3)

= p(B|A)p(A)

p(B|A)p(A) ⇒ p(A|B) = . p(B)

p(SNIa|binary) ' 10−12 yr−1 while p(binary|SNIa) = 1.

Conditional probability: p(A|B): probability of A, knowing B

Bayes’ rule (general probability theorem): By symmetry of A and B: p(A and B) = p(A|B)p(B)

Example of p(A|B) 6= p(B|A):

Prologue: Visual Representation of Bayes’ Rule (Venn Diagram)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 4 / 36 p(A and B) = . p(B)

⇔ probability of A, within B (new Universe): (2) p(A|B) = (2) + (3)

= p(B|A)p(A)

p(B|A)p(A) ⇒ p(A|B) = . p(B)

p(SNIa|binary) ' 10−12 yr−1 while p(binary|SNIa) = 1.

Conditional probability: p(A|B): probability of A, knowing B

Bayes’ rule (general probability theorem): By symmetry of A and B: p(A and B) = p(A|B)p(B)

Example of p(A|B) 6= p(B|A):

Prologue: Visual Representation of Bayes’ Rule (Venn Diagram)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 4 / 36 p(A and B) = . p(B)

⇔ probability of A, within B (new Universe): (2) p(A|B) = (2) + (3)

= p(B|A)p(A)

p(B|A)p(A) ⇒ p(A|B) = . p(B)

p(SNIa|binary) ' 10−12 yr−1 while p(binary|SNIa) = 1.

Conditional probability: p(A|B): probability of A, knowing B

Bayes’ rule (general probability theorem): By symmetry of A and B: p(A and B) = p(A|B)p(B)

Example of p(A|B) 6= p(B|A):

Prologue: Visual Representation of Bayes’ Rule (Venn Diagram)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 4 / 36 p(A and B) = . p(B)

⇔ probability of A, within B (new Universe): (2) p(A|B) = (2) + (3)

= p(B|A)p(A)

p(B|A)p(A) ⇒ p(A|B) = . p(B)

p(SNIa|binary) ' 10−12 yr−1 while p(binary|SNIa) = 1.

Conditional probability: p(A|B): probability of A, knowing B

Bayes’ rule (general probability theorem): By symmetry of A and B: p(A and B) = p(A|B)p(B)

Example of p(A|B) 6= p(B|A):

Prologue: Visual Representation of Bayes’ Rule (Venn Diagram)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 4 / 36 p(A and B) = . p(B)

= p(B|A)p(A)

p(B|A)p(A) ⇒ p(A|B) = . p(B)

p(SNIa|binary) ' 10−12 yr−1 while p(binary|SNIa) = 1.

⇔ probability of A, within B (new Universe): (2) p(A|B) = (2) + (3)

Bayes’ rule (general probability theorem): By symmetry of A and B: p(A and B) = p(A|B)p(B)

Example of p(A|B) 6= p(B|A):

Prologue: Visual Representation of Bayes’ Rule (Venn Diagram)

Conditional probability: p(A|B): probability of A, knowing B

F. Galliano (AIM) Astromind 2019, CEA/Saclay 4 / 36 p(A and B) = . p(B)

= p(B|A)p(A)

p(B|A)p(A) ⇒ p(A|B) = . p(B)

p(SNIa|binary) ' 10−12 yr−1 while p(binary|SNIa) = 1.

(2) p(A|B) = (2) + (3)

Bayes’ rule (general probability theorem): By symmetry of A and B: p(A and B) = p(A|B)p(B)

Example of p(A|B) 6= p(B|A):

Prologue: Visual Representation of Bayes’ Rule (Venn Diagram)

Conditional probability: p(A|B): probability of A, knowing B ⇔ probability of A, within B (new Universe):

F. Galliano (AIM) Astromind 2019, CEA/Saclay 4 / 36 = p(B|A)p(A)

p(B|A)p(A) ⇒ p(A|B) = . p(B)

p(SNIa|binary) ' 10−12 yr−1 while p(binary|SNIa) = 1.

p(A and B) = . p(B)

Bayes’ rule (general probability theorem): By symmetry of A and B: p(A and B) = p(A|B)p(B)

Example of p(A|B) 6= p(B|A):

Prologue: Visual Representation of Bayes’ Rule (Venn Diagram)

Conditional probability: p(A|B): probability of A, knowing B ⇔ probability of A, within B (new Universe): (2) p(A|B) = (2) + (3)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 4 / 36 = p(B|A)p(A)

p(B|A)p(A) ⇒ p(A|B) = . p(B)

p(SNIa|binary) ' 10−12 yr−1 while p(binary|SNIa) = 1.

Bayes’ rule (general probability theorem): By symmetry of A and B: p(A and B) = p(A|B)p(B)

Example of p(A|B) 6= p(B|A):

Prologue: Visual Representation of Bayes’ Rule (Venn Diagram)

Conditional probability: p(A|B): probability of A, knowing B ⇔ probability of A, within B (new Universe): (2) p(A and B) p(A|B) = = . (2) + (3) p(B)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 4 / 36 p(SNIa|binary) ' 10−12 yr−1 while p(binary|SNIa) = 1.

= p(B|A)p(A)

p(B|A)p(A) ⇒ p(A|B) = . p(B)

Example of p(A|B) 6= p(B|A):

Prologue: Visual Representation of Bayes’ Rule (Venn Diagram)

Conditional probability: p(A|B): probability of A, knowing B ⇔ probability of A, within B (new Universe): (2) p(A and B) p(A|B) = = . (2) + (3) p(B)

Bayes’ rule (general probability theorem): By symmetry of A and B: p(A and B) = p(A|B)p(B)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 4 / 36 p(SNIa|binary) ' 10−12 yr−1 while p(binary|SNIa) = 1.

p(B|A)p(A) ⇒ p(A|B) = . p(B)

Example of p(A|B) 6= p(B|A):

Prologue: Visual Representation of Bayes’ Rule (Venn Diagram)

Conditional probability: p(A|B): probability of A, knowing B ⇔ probability of A, within B (new Universe): (2) p(A and B) p(A|B) = = . (2) + (3) p(B)

Bayes’ rule (general probability theorem): By symmetry of A and B: p(A and B) = p(A|B)p(B) = p(B|A)p(A)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 4 / 36 p(SNIa|binary) ' 10−12 yr−1 while p(binary|SNIa) = 1.

Example of p(A|B) 6= p(B|A):

Prologue: Visual Representation of Bayes’ Rule (Venn Diagram)

Conditional probability: p(A|B): probability of A, knowing B ⇔ probability of A, within B (new Universe): (2) p(A and B) p(A|B) = = . (2) + (3) p(B)

Bayes’ rule (general probability theorem): By symmetry of A and B: p(A and B) = p(A|B)p(B) = p(B|A)p(A)

p(B|A)p(A) ⇒ p(A|B) = . p(B)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 4 / 36 p(SNIa|binary) ' 10−12 yr−1 while p(binary|SNIa) = 1.

Prologue: Visual Representation of Bayes’ Rule (Venn Diagram)

Conditional probability: p(A|B): probability of A, knowing B ⇔ probability of A, within B (new Universe): (2) p(A and B) p(A|B) = = . (2) + (3) p(B)

Bayes’ rule (general probability theorem): By symmetry of A and B: p(A and B) = p(A|B)p(B) = p(B|A)p(A)

p(B|A)p(A) ⇒ p(A|B) = . p(B)

Example of p(A|B) 6= p(B|A):

F. Galliano (AIM) Astromind 2019, CEA/Saclay 4 / 36 while p(binary|SNIa) = 1.

Prologue: Visual Representation of Bayes’ Rule (Venn Diagram)

Conditional probability: p(A|B): probability of A, knowing B ⇔ probability of A, within B (new Universe): (2) p(A and B) p(A|B) = = . (2) + (3) p(B)

Bayes’ rule (general probability theorem): By symmetry of A and B: p(A and B) = p(A|B)p(B) = p(B|A)p(A)

p(B|A)p(A) ⇒ p(A|B) = . p(B)

Example of p(A|B) 6= p(B|A):

p(SNIa|binary) ' 10−12 yr−1

F. Galliano (AIM) Astromind 2019, CEA/Saclay 4 / 36 Prologue: Visual Representation of Bayes’ Rule (Venn Diagram)

Conditional probability: p(A|B): probability of A, knowing B ⇔ probability of A, within B (new Universe): (2) p(A and B) p(A|B) = = . (2) + (3) p(B)

Bayes’ rule (general probability theorem): By symmetry of A and B: p(A and B) = p(A|B)p(B) = p(B|A)p(A)

p(B|A)p(A) ⇒ p(A|B) = . p(B)

Example of p(A|B) 6= p(B|A):

p(SNIa|binary) ' 10−12 yr−1 while p(binary|SNIa) = 1.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 4 / 36 Probability: quantiﬁcation of the Probability: limit to inﬁnity of the plausibility of a proposition, occurence frequency of an with incomplete knowledge event, in a sequence of (subjective point of view). repeated experiments. ⇒ probabilities can be assigned to parameters ⇒ probabilities can not be assigned to and hypotheses. parameters or hypotheses. Inference: sample the distribution of Inference: sample the likelihood, parameters, conditional on conditional on the the data (Bayes’ rule): parameters: p(par|data) ∝ p(data|par) × p(par) . L(par) ≡ p(data|par). | {z } | {z } | {z } posterior likelihood prior Results: describe how the derived Results: the posterior contains all the parameter value would vary if relevant information. we were to repeat the ⇒ use it to give credible ranges, test experiment in the same hypotheses, compare models, etc. conditions.

The Bayesian View The Frequentist View There is an objective truth, but our knowledge There is an objective truth, and science should is only partial and subjective. not deal with subjective notions.

Two Conceptions of Probability & Uncertainty

F. Galliano (AIM) Astromind 2019, CEA/Saclay 5 / 36 Probability: limit to inﬁnity of the occurence frequency of an event, in a sequence of repeated experiments. ⇒ probabilities can not be assigned to parameters or hypotheses. Inference: sample the likelihood, conditional on the parameters: L(par) ≡ p(data|par).

Results: describe how the derived parameter value would vary if we were to repeat the experiment in the same conditions.

The Frequentist View There is an objective truth, and science should not deal with subjective notions. Probability: quantiﬁcation of the plausibility of a proposition, with incomplete knowledge (subjective point of view). ⇒ probabilities can be assigned to parameters and hypotheses. Inference: sample the distribution of parameters, conditional on the data (Bayes’ rule): p(par|data) ∝ p(data|par) × p(par) . | {z } | {z } | {z } posterior likelihood prior

Results: the posterior contains all the relevant information. ⇒ use it to give credible ranges, test hypotheses, compare models, etc.

Two Conceptions of Probability & Uncertainty

The Bayesian View There is an objective truth, but our knowledge is only partial and subjective.

Results: describe how the derived parameter value would vary if we were to repeat the experiment in the same conditions.

The Frequentist View There is an objective truth, and science should not deal with subjective notions.

⇒ probabilities can be assigned to parameters and hypotheses. Inference: sample the distribution of parameters, conditional on the data (Bayes’ rule): p(par|data) ∝ p(data|par) × p(par) . | {z } | {z } | {z } posterior likelihood prior

Results: the posterior contains all the relevant information. ⇒ use it to give credible ranges, test hypotheses, compare models, etc.

Two Conceptions of Probability & Uncertainty

Results: describe how the derived parameter value would vary if we were to repeat the experiment in the same conditions.

The Frequentist View There is an objective truth, and science should not deal with subjective notions.

Inference: sample the distribution of parameters, conditional on the data (Bayes’ rule): p(par|data) ∝ p(data|par) × p(par) . | {z } | {z } | {z } posterior likelihood prior

Results: the posterior contains all the relevant information. ⇒ use it to give credible ranges, test hypotheses, compare models, etc.

Two Conceptions of Probability & Uncertainty

Results: describe how the derived parameter value would vary if we were to repeat the experiment in the same conditions.

The Frequentist View There is an objective truth, and science should not deal with subjective notions.

p(par|data) ∝ p(data|par) × p(par) . | {z } | {z } | {z } posterior likelihood prior

Results: the posterior contains all the relevant information. ⇒ use it to give credible ranges, test hypotheses, compare models, etc.

Two Conceptions of Probability & Uncertainty

The Bayesian View There is an objective truth, but our knowledge is only partial and subjective. Probability: quantiﬁcation of the plausibility of a proposition, with incomplete knowledge (subjective point of view). ⇒ probabilities can be assigned to parameters and hypotheses. Inference: sample the distribution of parameters, conditional on the data (Bayes’ rule):

Results: describe how the derived parameter value would vary if we were to repeat the experiment in the same conditions.

The Frequentist View There is an objective truth, and science should not deal with subjective notions.

∝ p(data|par) × p(par) . | {z } | {z } likelihood prior

Results: the posterior contains all the relevant information. ⇒ use it to give credible ranges, test hypotheses, compare models, etc.

Two Conceptions of Probability & Uncertainty

Results: describe how the derived parameter value would vary if we were to repeat the experiment in the same conditions.

The Frequentist View There is an objective truth, and science should not deal with subjective notions.

p(par) . | {z } prior

Results: the posterior contains all the relevant information. ⇒ use it to give credible ranges, test hypotheses, compare models, etc.

Two Conceptions of Probability & Uncertainty

Results: describe how the derived parameter value would vary if we were to repeat the experiment in the same conditions.

The Frequentist View There is an objective truth, and science should not deal with subjective notions.

Results: the posterior contains all the relevant information. ⇒ use it to give credible ranges, test hypotheses, compare models, etc.

Two Conceptions of Probability & Uncertainty

Results: describe how the derived parameter value would vary if we were to repeat the experiment in the same conditions.

The Frequentist View There is an objective truth, and science should not deal with subjective notions.

⇒ use it to give credible ranges, test hypotheses, compare models, etc.

Two Conceptions of Probability & Uncertainty

Results: the posterior contains all the relevant information.

Results: describe how the derived parameter value would vary if we were to repeat the experiment in the same conditions.

The Frequentist View There is an objective truth, and science should not deal with subjective notions.

Two Conceptions of Probability & Uncertainty

Results: the posterior contains all the relevant information. ⇒ use it to give credible ranges, test hypotheses, compare models, etc.

Results: describe how the derived parameter value would vary if we were to repeat the experiment in the same conditions.

Two Conceptions of Probability & Uncertainty

The Bayesian View The Frequentist View There is an objective truth, but our knowledge There is an objective truth, and science should is only partial and subjective. not deal with subjective notions. Probability: quantiﬁcation of the plausibility of a proposition, with incomplete knowledge (subjective point of view). ⇒ probabilities can be assigned to parameters and hypotheses. Inference: sample the distribution of parameters, conditional on the data (Bayes’ rule): p(par|data) ∝ p(data|par) × p(par) . | {z } | {z } | {z } posterior likelihood prior

Results: the posterior contains all the relevant information. ⇒ use it to give credible ranges, test hypotheses, compare models, etc.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 5 / 36 ⇒ probabilities can not be assigned to parameters or hypotheses. Inference: sample the likelihood, conditional on the parameters: L(par) ≡ p(data|par).

Results: describe how the derived parameter value would vary if we were to repeat the experiment in the same conditions.

Two Conceptions of Probability & Uncertainty

Results: the posterior contains all the relevant information. ⇒ use it to give credible ranges, test hypotheses, compare models, etc.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 5 / 36 Inference: sample the likelihood, conditional on the parameters: L(par) ≡ p(data|par).

Results: describe how the derived parameter value would vary if we were to repeat the experiment in the same conditions.

Two Conceptions of Probability & Uncertainty

Results: the posterior contains all the relevant information. ⇒ use it to give credible ranges, test hypotheses, compare models, etc.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 5 / 36 L(par) ≡ p(data|par).

Results: describe how the derived parameter value would vary if we were to repeat the experiment in the same conditions.

Two Conceptions of Probability & Uncertainty

The Bayesian View The Frequentist View There is an objective truth, but our knowledge There is an objective truth, and science should is only partial and subjective. not deal with subjective notions. Probability: quantiﬁcation of the Probability: limit to inﬁnity of the plausibility of a proposition, occurence frequency of an with incomplete knowledge event, in a sequence of (subjective point of view). repeated experiments. ⇒ probabilities can be assigned to parameters ⇒ probabilities can not be assigned to and hypotheses. parameters or hypotheses. Inference: sample the distribution of Inference: sample the likelihood, parameters, conditional on conditional on the the data (Bayes’ rule): parameters: p(par|data) ∝ p(data|par) × p(par) . | {z } | {z } | {z } posterior likelihood prior

Results: the posterior contains all the relevant information. ⇒ use it to give credible ranges, test hypotheses, compare models, etc.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 5 / 36 Results: describe how the derived parameter value would vary if we were to repeat the experiment in the same conditions.

Two Conceptions of Probability & Uncertainty

The Bayesian View The Frequentist View There is an objective truth, but our knowledge There is an objective truth, and science should is only partial and subjective. not deal with subjective notions. Probability: quantiﬁcation of the Probability: limit to inﬁnity of the plausibility of a proposition, occurence frequency of an with incomplete knowledge event, in a sequence of (subjective point of view). repeated experiments. ⇒ probabilities can be assigned to parameters ⇒ probabilities can not be assigned to and hypotheses. parameters or hypotheses. Inference: sample the distribution of Inference: sample the likelihood, parameters, conditional on conditional on the the data (Bayes’ rule): parameters: p(par|data) ∝ p(data|par) × p(par) . L(par) ≡ p(data|par). | {z } | {z } | {z } posterior likelihood prior

Results: the posterior contains all the relevant information. ⇒ use it to give credible ranges, test hypotheses, compare models, etc.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 5 / 36 Two Conceptions of Probability & Uncertainty

The Bayesian View The Frequentist View There is an objective truth, but our knowledge There is an objective truth, and science should is only partial and subjective. not deal with subjective notions. Probability: quantiﬁcation of the Probability: limit to inﬁnity of the plausibility of a proposition, occurence frequency of an with incomplete knowledge event, in a sequence of (subjective point of view). repeated experiments. ⇒ probabilities can be assigned to parameters ⇒ probabilities can not be assigned to and hypotheses. parameters or hypotheses. Inference: sample the distribution of Inference: sample the likelihood, parameters, conditional on conditional on the the data (Bayes’ rule): parameters: p(par|data) ∝ p(data|par) × p(par) . L(par) ≡ p(data|par). | {z } | {z } | {z } posterior likelihood prior Results: describe how the derived Results: the posterior contains all the parameter value would vary if relevant information. we were to repeat the ⇒ use it to give credible ranges, test experiment in the same hypotheses, compare models, etc. conditions.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 5 / 36 = 53.9 ± 8.1.

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F i=1 i σF F? ' ± √ N N

1 |{z} 2 Sampling Fi ⇒ conﬁdence interval. prior Bootstrapping: MCMC: F? ' 53.9 ± 8.1; F? ' 53.9 ± 8.1; 95 % conﬁdence 95 % credible interval: range: [37.8, 69.8]. [37.8, 69.8].

Observations

True ﬂux, F? = 42;

Bayesian Solution: Frequentist Solution:

2 1 Maximum-likelihood, F ; Y (F? − Fi ) ML p(F?|Fi ) ∝ exp − × 2σ2 i F

Comparison of the Two Approaches on a Simple Case

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 = 53.9 ± 8.1.

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F i=1 i σF F? ' ± √ N N

1 |{z} 2 Sampling Fi ⇒ conﬁdence interval. prior Bootstrapping: MCMC: F? ' 53.9 ± 8.1; F? ' 53.9 ± 8.1; 95 % conﬁdence 95 % credible interval: range: [37.8, 69.8]. [37.8, 69.8].

Observations

True ﬂux, F? = 42;

Bayesian Solution: Frequentist Solution:

2 1 Maximum-likelihood, F ; Y (F? − Fi ) ML p(F?|Fi ) ∝ exp − × 2σ2 i F

Comparison of the Two Approaches on a Simple Case

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 = 53.9 ± 8.1.

1 |{z} 2 Sampling Fi ⇒ conﬁdence interval. prior Bootstrapping: MCMC: F? ' 53.9 ± 8.1; F? ' 53.9 ± 8.1; 95 % conﬁdence 95 % credible interval: range: [37.8, 69.8]. [37.8, 69.8].

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F i=1 i σF F? ' ± √ N N

Bayesian Solution: Frequentist Solution:

2 1 Maximum-likelihood, F ; Y (F? − Fi ) ML p(F?|Fi ) ∝ exp − × 2σ2 i F

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 = 53.9 ± 8.1.

1 |{z} 2 Sampling Fi ⇒ conﬁdence interval. prior Bootstrapping: MCMC: F? ' 53.9 ± 8.1; F? ' 53.9 ± 8.1; 95 % conﬁdence 95 % credible interval: range: [37.8, 69.8]. [37.8, 69.8].

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F i=1 i σF F? ' ± √ N N

Bayesian Solution: Frequentist Solution:

2 1 Maximum-likelihood, F ; Y (F? − Fi ) ML p(F?|Fi ) ∝ exp − × 2σ2 i F

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42; Observation Number

Measured Flux

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 = 53.9 ± 8.1.

1 |{z} 2 Sampling Fi ⇒ conﬁdence interval. prior Bootstrapping: MCMC: F? ' 53.9 ± 8.1; F? ' 53.9 ± 8.1; 95 % conﬁdence 95 % credible interval: range: [37.8, 69.8]. [37.8, 69.8].

Gaussian noise, σF = 14. ⇒ classic solution: PN F i=1 i σF F? ' ± √ N N

Bayesian Solution: Frequentist Solution:

2 1 Maximum-likelihood, F ; Y (F? − Fi ) ML p(F?|Fi ) ∝ exp − × 2σ2 i F

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ; Observation Number

Measured Flux

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 = 53.9 ± 8.1.

1 |{z} 2 Sampling Fi ⇒ conﬁdence interval. prior Bootstrapping: MCMC: F? ' 53.9 ± 8.1; F? ' 53.9 ± 8.1; 95 % conﬁdence 95 % credible interval: range: [37.8, 69.8]. [37.8, 69.8].

⇒ classic solution: PN F i=1 i σF F? ' ± √ N N

Bayesian Solution: Frequentist Solution:

2 1 Maximum-likelihood, F ; Y (F? − Fi ) ML p(F?|Fi ) ∝ exp − × 2σ2 i F

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. Observation Number

Measured Flux

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 = 53.9 ± 8.1.

1 |{z} 2 Sampling Fi ⇒ conﬁdence interval. prior Bootstrapping: MCMC: F? ' 53.9 ± 8.1; F? ' 53.9 ± 8.1; 95 % conﬁdence 95 % credible interval: range: [37.8, 69.8]. [37.8, 69.8].

⇒ classic solution: PN F i=1 i σF F? ' ± √ N N

Bayesian Solution: Frequentist Solution:

2 1 Maximum-likelihood, F ; Y (F? − Fi ) ML p(F?|Fi ) ∝ exp − × 2σ2 i F

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. Observation Number

Measured Flux

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 1 |{z} 2 Sampling Fi ⇒ conﬁdence interval. prior Bootstrapping: MCMC: F? ' 53.9 ± 8.1; F? ' 53.9 ± 8.1; 95 % conﬁdence 95 % credible interval: range: [37.8, 69.8]. [37.8, 69.8].

= 53.9 ± 8.1.

Bayesian Solution: Frequentist Solution:

2 1 Maximum-likelihood, F ; Y (F? − Fi ) ML p(F?|Fi ) ∝ exp − × 2σ2 i F

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ N N Measured Flux

Bayesian Solution: Frequentist Solution:

2 1 Maximum-likelihood, F ; Y (F? − Fi ) ML p(F?|Fi ) ∝ exp − × 2σ2 i F

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 2 Sampling Fi ⇒ conﬁdence interval.

Bootstrapping:

F? ' 53.9 ± 8.1; 95 % conﬁdence interval: [37.8, 69.8].

Frequentist Solution:

1 Maximum-likelihood, FML; 1 |{z} prior MCMC:

F? ' 53.9 ± 8.1; 95 % credible range: [37.8, 69.8].

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

Bayesian Solution:

2 Y (F? − Fi ) p(F?|Fi ) ∝ exp − × 2σ2 i F

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 2 Sampling Fi ⇒ conﬁdence interval.

Bootstrapping:

F? ' 53.9 ± 8.1; 95 % conﬁdence interval: [37.8, 69.8].

Frequentist Solution:

1 Maximum-likelihood, FML;

MCMC:

F? ' 53.9 ± 8.1; 95 % credible range: [37.8, 69.8].

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

Bayesian Solution:

2 Y (F? − Fi ) p(F?|Fi ) ∝ exp − 2 × 1 2σF |{z} i prior

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 2 Sampling Fi ⇒ conﬁdence interval.

Bootstrapping:

F? ' 53.9 ± 8.1; 95 % conﬁdence interval: [37.8, 69.8].

Frequentist Solution:

1 Maximum-likelihood, FML;

MCMC:

F? ' 53.9 ± 8.1; 95 % credible range: [37.8, 69.8].

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

Bayesian Solution:

2 Y (F? − Fi ) p(F?|Fi ) ∝ exp − 2 × 1 2σF |{z} i prior

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 2 Sampling Fi ⇒ conﬁdence interval.

Bootstrapping:

F? ' 53.9 ± 8.1; 95 % conﬁdence interval: [37.8, 69.8].

Frequentist Solution:

1 Maximum-likelihood, FML;

MCMC:

F? ' 53.9 ± 8.1; 95 % credible range: [37.8, 69.8].

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

Bayesian Solution:

2 Y (F? − Fi ) p(F?|Fi ) ∝ exp − 2 × 1 2σF |{z} i prior

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 2 Sampling Fi ⇒ conﬁdence interval.

Bootstrapping:

F? ' 53.9 ± 8.1; 95 % conﬁdence interval: [37.8, 69.8].

Frequentist Solution:

1 Maximum-likelihood, FML;

MCMC:

F? ' 53.9 ± 8.1; 95 % credible range: [37.8, 69.8].

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

Bayesian Solution:

2 Y (F? − Fi ) p(F?|Fi ) ∝ exp − 2 × 1 2σF |{z} i prior

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 2 Sampling Fi ⇒ conﬁdence interval.

Bootstrapping:

F? ' 53.9 ± 8.1; 95 % conﬁdence interval: [37.8, 69.8].

Frequentist Solution:

1 Maximum-likelihood, FML;

MCMC:

F? ' 53.9 ± 8.1; 95 % credible range: [37.8, 69.8].

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

Bayesian Solution:

2 Y (F? − Fi ) p(F?|Fi ) ∝ exp − 2 × 1 2σF |{z} i prior

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 2 Sampling Fi ⇒ conﬁdence interval.

Bootstrapping:

F? ' 53.9 ± 8.1; 95 % conﬁdence interval: [37.8, 69.8].

Frequentist Solution:

1 Maximum-likelihood, FML;

MCMC:

F? ' 53.9 ± 8.1; 95 % credible range: [37.8, 69.8].

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

Bayesian Solution:

2 Y (F? − Fi ) p(F?|Fi ) ∝ exp − 2 × 1 2σF |{z} i prior

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 2 Sampling Fi ⇒ conﬁdence interval.

Bootstrapping:

F? ' 53.9 ± 8.1; 95 % conﬁdence interval: [37.8, 69.8].

Frequentist Solution:

1 Maximum-likelihood, FML;

MCMC:

F? ' 53.9 ± 8.1; 95 % credible range: [37.8, 69.8].

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

Bayesian Solution:

2 Y (F? − Fi ) p(F?|Fi ) ∝ exp − 2 × 1 2σF |{z} i prior

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 2 Sampling Fi ⇒ conﬁdence interval.

Bootstrapping:

F? ' 53.9 ± 8.1; 95 % conﬁdence interval: [37.8, 69.8].

Frequentist Solution:

1 Maximum-likelihood, FML;

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

Bayesian Solution:

2 Y (F? − Fi ) p(F?|Fi ) ∝ exp − 2 × 1 2σF |{z} i prior

MCMC:

F? ' 53.9 ± 8.1; 95 % credible range: [37.8, 69.8].

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 2 Sampling Fi ⇒ conﬁdence interval.

Bootstrapping:

F? ' 53.9 ± 8.1; 95 % conﬁdence interval: [37.8, 69.8].

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

Bayesian Solution: Frequentist Solution:

2 1 Maximum-likelihood, F ; Y (F? − Fi ) ML p(F?|Fi ) ∝ exp − 2 × 1 2σF |{z} i prior

MCMC:

F? ' 53.9 ± 8.1; 95 % credible range: [37.8, 69.8].

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 Bootstrapping:

F? ' 53.9 ± 8.1; 95 % conﬁdence interval: [37.8, 69.8].

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

Bayesian Solution: Frequentist Solution:

2 1 Maximum-likelihood, F ; Y (F? − Fi ) ML p(F?|Fi ) ∝ exp − × 1 2 2 Sampling Fi ⇒ conﬁdence interval. 2σF |{z} i prior

MCMC:

F? ' 53.9 ± 8.1; 95 % credible range: [37.8, 69.8].

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 Bootstrapping:

F? ' 53.9 ± 8.1; 95 % conﬁdence interval: [37.8, 69.8].

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

Bayesian Solution: Frequentist Solution:

2 1 Maximum-likelihood, F ; Y (F? − Fi ) ML p(F?|Fi ) ∝ exp − × 1 2 2 Sampling Fi ⇒ conﬁdence interval. 2σF |{z} i prior

MCMC:

F? ' 53.9 ± 8.1; 95 % credible range: [37.8, 69.8].

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 Bootstrapping:

F? ' 53.9 ± 8.1; 95 % conﬁdence interval: [37.8, 69.8].

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

Bayesian Solution: Frequentist Solution:

2 1 Maximum-likelihood, F ; Y (F? − Fi ) ML p(F?|Fi ) ∝ exp − × 1 2 2 Sampling Fi ⇒ conﬁdence interval. 2σF |{z} i prior

MCMC:

F? ' 53.9 ± 8.1; 95 % credible range: [37.8, 69.8].

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 Bootstrapping:

F? ' 53.9 ± 8.1; 95 % conﬁdence interval: [37.8, 69.8].

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

Bayesian Solution: Frequentist Solution:

2 1 Maximum-likelihood, F ; Y (F? − Fi ) ML p(F?|Fi ) ∝ exp − × 1 2 2 Sampling Fi ⇒ conﬁdence interval. 2σF |{z} i prior

MCMC:

F? ' 53.9 ± 8.1; 95 % credible range: [37.8, 69.8].

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 Bootstrapping:

F? ' 53.9 ± 8.1; 95 % conﬁdence interval: [37.8, 69.8].

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

Bayesian Solution: Frequentist Solution:

2 1 Maximum-likelihood, F ; Y (F? − Fi ) ML p(F?|Fi ) ∝ exp − × 1 2 2 Sampling Fi ⇒ conﬁdence interval. 2σF |{z} i prior

MCMC:

F? ' 53.9 ± 8.1; 95 % credible range: [37.8, 69.8].

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 Bootstrapping:

F? ' 53.9 ± 8.1; 95 % conﬁdence interval: [37.8, 69.8].

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

Bayesian Solution: Frequentist Solution:

2 1 Maximum-likelihood, F ; Y (F? − Fi ) ML p(F?|Fi ) ∝ exp − × 1 2 2 Sampling Fi ⇒ conﬁdence interval. 2σF |{z} i prior

MCMC:

F? ' 53.9 ± 8.1; 95 % credible range: [37.8, 69.8].

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 Bootstrapping:

F? ' 53.9 ± 8.1; 95 % conﬁdence interval: [37.8, 69.8].

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

Bayesian Solution: Frequentist Solution:

2 1 Maximum-likelihood, F ; Y (F? − Fi ) ML p(F?|Fi ) ∝ exp − × 1 2 2 Sampling Fi ⇒ conﬁdence interval. 2σF |{z} i prior

MCMC:

F? ' 53.9 ± 8.1; 95 % credible range: [37.8, 69.8].

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 Bootstrapping:

F? ' 53.9 ± 8.1; 95 % conﬁdence interval: [37.8, 69.8].

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

Bayesian Solution: Frequentist Solution:

2 1 Maximum-likelihood, F ; Y (F? − Fi ) ML p(F?|Fi ) ∝ exp − × 1 2 2 Sampling Fi ⇒ conﬁdence interval. 2σF |{z} i prior

MCMC:

F? ' 53.9 ± 8.1; 95 % credible range: [37.8, 69.8].

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 Bootstrapping:

F? ' 53.9 ± 8.1; 95 % conﬁdence interval: [37.8, 69.8].

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

Bayesian Solution: Frequentist Solution:

2 1 Maximum-likelihood, F ; Y (F? − Fi ) ML p(F?|Fi ) ∝ exp − × 1 2 2 Sampling Fi ⇒ conﬁdence interval. 2σF |{z} i prior

MCMC:

F? ' 53.9 ± 8.1; 95 % credible range: [37.8, 69.8].

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 Bootstrapping:

F? ' 53.9 ± 8.1; 95 % conﬁdence interval: [37.8, 69.8].

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

Bayesian Solution: Frequentist Solution:

2 1 Maximum-likelihood, F ; Y (F? − Fi ) ML p(F?|Fi ) ∝ exp − × 1 2 2 Sampling Fi ⇒ conﬁdence interval. 2σF |{z} i prior

MCMC:

F? ' 53.9 ± 8.1; 95 % credible range: [37.8, 69.8].

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 Bootstrapping:

F? ' 53.9 ± 8.1; 95 % conﬁdence interval: [37.8, 69.8].

Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

Bayesian Solution: Frequentist Solution:

2 1 Maximum-likelihood, F ; Y (F? − Fi ) ML p(F?|Fi ) ∝ exp − × 1 2 2 Sampling Fi ⇒ conﬁdence interval. 2σF |{z} i prior

MCMC:

F? ' 53.9 ± 8.1; 95 % credible range: [37.8, 69.8].

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 Comparison of the Two Approaches on a Simple Case

Observations

True ﬂux, F? = 42;

N = 3 observations, Fi ;

Gaussian noise, σF = 14. ⇒ classic solution: PN F Observation Number i=1 i σF F? ' ± √ = 53.9 ± 8.1. N N Measured Flux

Bayesian Solution: Frequentist Solution:

2 1 Maximum-likelihood, F ; Y (F? − Fi ) ML p(F?|Fi ) ∝ exp − × 1 2 2 Sampling Fi ⇒ conﬁdence interval. 2σF |{z} i prior

MCMC: Bootstrapping:

F? ' 53.9 ± 8.1; F? ' 53.9 ± 8.1; 95 % credible 95 % conﬁdence range: interval: [37.8, 69.8]. [37.8, 69.8].

F. Galliano (AIM) Astromind 2019, CEA/Saclay 6 / 36 (1) (N?) p(F?) ∝ 1 × p(F? |F?) × ... × p(F? |F?) |{z} |{z} | {z } new prior initial prior cumulated data

Example: the star is in a globular cluster& you know the distance. Prior: cluster mass function& mass/luminosity relation ⇒ luminosity function.

True ﬂux: F? = 42

Frequentist: F? ' 53.9 ± 8.1

Bayesian: F? ' 47.1 ± 8.0

Cumulation of data: if you perform a series of observations in this cluster:

Hierarchical model: consistently perform this process on all the data, at the same time.

Accounting for Additional Information:

Reﬁning the Prior Based on the Observations:

Diﬀerence Between the Two Approaches: Using an Informative Prior

F. Galliano (AIM) Astromind 2019, CEA/Saclay 7 / 36 (1) (N?) p(F?) ∝ 1 × p(F? |F?) × ... × p(F? |F?) |{z} |{z} | {z } new prior initial prior cumulated data

Cumulation of data: if you perform a series of observations in this cluster:

Hierarchical model: consistently perform this process on all the data, at the same time.

Example: the star is in a globular cluster& you know the distance. Prior: cluster mass function& mass/luminosity relation ⇒ luminosity function.

True ﬂux: F? = 42

Frequentist: F? ' 53.9 ± 8.1

Bayesian: F? ' 47.1 ± 8.0

Reﬁning the Prior Based on the Observations:

Diﬀerence Between the Two Approaches: Using an Informative Prior

Accounting for Additional Information:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 7 / 36 (1) (N?) p(F?) ∝ 1 × p(F? |F?) × ... × p(F? |F?) |{z} |{z} | {z } new prior initial prior cumulated data

Cumulation of data: if you perform a series of observations in this cluster:

Hierarchical model: consistently perform this process on all the data, at the same time.

Prior: cluster mass function& mass/luminosity relation ⇒ luminosity function.

True ﬂux: F? = 42

Frequentist: F? ' 53.9 ± 8.1

Bayesian: F? ' 47.1 ± 8.0

Reﬁning the Prior Based on the Observations:

Diﬀerence Between the Two Approaches: Using an Informative Prior

Accounting for Additional Information: Example: the star is in a globular cluster& you know the distance.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 7 / 36 (1) (N?) p(F?) ∝ 1 × p(F? |F?) × ... × p(F? |F?) |{z} |{z} | {z } new prior initial prior cumulated data

Cumulation of data: if you perform a series of observations in this cluster:

Hierarchical model: consistently perform this process on all the data, at the same time.

Prior: cluster mass function& mass/luminosity relation ⇒ luminosity function.

True ﬂux: F? = 42

Frequentist: F? ' 53.9 ± 8.1

Bayesian: F? ' 47.1 ± 8.0

Reﬁning the Prior Based on the Observations:

Diﬀerence Between the Two Approaches: Using an Informative Prior

Accounting for Additional Information: Example: the star is in a globular cluster& you know the distance.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 7 / 36 (1) (N?) p(F?) ∝ 1 × p(F? |F?) × ... × p(F? |F?) |{z} |{z} | {z } new prior initial prior cumulated data

Cumulation of data: if you perform a series of observations in this cluster:

Hierarchical model: consistently perform this process on all the data, at the same time.

Prior: cluster mass function& mass/luminosity relation ⇒ luminosity function.

True ﬂux: F? = 42

Frequentist: F? ' 53.9 ± 8.1

Bayesian: F? ' 47.1 ± 8.0

Reﬁning the Prior Based on the Observations:

Diﬀerence Between the Two Approaches: Using an Informative Prior

Accounting for Additional Information: Example: the star is in a globular cluster& you know the distance.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 7 / 36 (1) (N?) p(F?) ∝ 1 × p(F? |F?) × ... × p(F? |F?) |{z} |{z} | {z } new prior initial prior cumulated data

Cumulation of data: if you perform a series of observations in this cluster:

Hierarchical model: consistently perform this process on all the data, at the same time.

True ﬂux: F? = 42

Frequentist: F? ' 53.9 ± 8.1

Bayesian: F? ' 47.1 ± 8.0

Reﬁning the Prior Based on the Observations:

Diﬀerence Between the Two Approaches: Using an Informative Prior

Accounting for Additional Information: Example: the star is in a globular cluster& you know the distance. Prior: cluster mass function& mass/luminosity relation ⇒ luminosity function.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 7 / 36 (1) (N?) p(F?) ∝ 1 × p(F? |F?) × ... × p(F? |F?) |{z} |{z} | {z } new prior initial prior cumulated data

Cumulation of data: if you perform a series of observations in this cluster:

Hierarchical model: consistently perform this process on all the data, at the same time.

Frequentist: F? ' 53.9 ± 8.1

Bayesian: F? ' 47.1 ± 8.0

Reﬁning the Prior Based on the Observations:

Diﬀerence Between the Two Approaches: Using an Informative Prior

Accounting for Additional Information: Example: the star is in a globular cluster& you know the distance. Prior: cluster mass function& mass/luminosity relation ⇒ luminosity function.

True ﬂux: F? = 42

F. Galliano (AIM) Astromind 2019, CEA/Saclay 7 / 36 (1) (N?) p(F?) ∝ 1 × p(F? |F?) × ... × p(F? |F?) |{z} |{z} | {z } new prior initial prior cumulated data

Cumulation of data: if you perform a series of observations in this cluster:

Hierarchical model: consistently perform this process on all the data, at the same time.

Frequentist: F? ' 53.9 ± 8.1

Bayesian: F? ' 47.1 ± 8.0

Reﬁning the Prior Based on the Observations:

Diﬀerence Between the Two Approaches: Using an Informative Prior

Accounting for Additional Information: Example: the star is in a globular cluster& you know the distance. Prior: cluster mass function& mass/luminosity relation ⇒ luminosity function.

True ﬂux: F? = 42

F. Galliano (AIM) Astromind 2019, CEA/Saclay 7 / 36 (1) (N?) p(F?) ∝ 1 × p(F? |F?) × ... × p(F? |F?) |{z} |{z} | {z } new prior initial prior cumulated data

Cumulation of data: if you perform a series of observations in this cluster:

Hierarchical model: consistently perform this process on all the data, at the same time.

Bayesian: F? ' 47.1 ± 8.0

Reﬁning the Prior Based on the Observations:

Diﬀerence Between the Two Approaches: Using an Informative Prior

Accounting for Additional Information: Example: the star is in a globular cluster& you know the distance. Prior: cluster mass function& mass/luminosity relation ⇒ luminosity function.

True ﬂux: F? = 42

Frequentist: F? ' 53.9 ± 8.1

F. Galliano (AIM) Astromind 2019, CEA/Saclay 7 / 36 (1) (N?) p(F?) ∝ 1 × p(F? |F?) × ... × p(F? |F?) |{z} |{z} | {z } new prior initial prior cumulated data

Cumulation of data: if you perform a series of observations in this cluster:

Hierarchical model: consistently perform this process on all the data, at the same time.

Bayesian: F? ' 47.1 ± 8.0

Reﬁning the Prior Based on the Observations:

Diﬀerence Between the Two Approaches: Using an Informative Prior

Accounting for Additional Information: Example: the star is in a globular cluster& you know the distance. Prior: cluster mass function& mass/luminosity relation ⇒ luminosity function.

True ﬂux: F? = 42

Frequentist: F? ' 53.9 ± 8.1

F. Galliano (AIM) Astromind 2019, CEA/Saclay 7 / 36 (1) (N?) p(F?) ∝ 1 × p(F? |F?) × ... × p(F? |F?) |{z} |{z} | {z } new prior initial prior cumulated data

Cumulation of data: if you perform a series of observations in this cluster:

Hierarchical model: consistently perform this process on all the data, at the same time.

Bayesian: F? ' 47.1 ± 8.0

Reﬁning the Prior Based on the Observations:

Diﬀerence Between the Two Approaches: Using an Informative Prior

Accounting for Additional Information: Example: the star is in a globular cluster& you know the distance. Prior: cluster mass function& mass/luminosity relation ⇒ luminosity function.

True ﬂux: F? = 42

Frequentist: F? ' 53.9 ± 8.1

F. Galliano (AIM) Astromind 2019, CEA/Saclay 7 / 36 (1) (N?) p(F?) ∝ 1 × p(F? |F?) × ... × p(F? |F?) |{z} |{z} | {z } new prior initial prior cumulated data

Cumulation of data: if you perform a series of observations in this cluster:

Hierarchical model: consistently perform this process on all the data, at the same time.

Reﬁning the Prior Based on the Observations:

Diﬀerence Between the Two Approaches: Using an Informative Prior

Accounting for Additional Information: Example: the star is in a globular cluster& you know the distance. Prior: cluster mass function& mass/luminosity relation ⇒ luminosity function.

True ﬂux: F? = 42

Frequentist: F? ' 53.9 ± 8.1

Bayesian: F? ' 47.1 ± 8.0

F. Galliano (AIM) Astromind 2019, CEA/Saclay 7 / 36 (1) (N?) p(F?) ∝ 1 × p(F? |F?) × ... × p(F? |F?) |{z} |{z} | {z } new prior initial prior cumulated data

Cumulation of data: if you perform a series of observations in this cluster:

Hierarchical model: consistently perform this process on all the data, at the same time.

Diﬀerence Between the Two Approaches: Using an Informative Prior

Accounting for Additional Information: Example: the star is in a globular cluster& you know the distance. Prior: cluster mass function& mass/luminosity relation ⇒ luminosity function.

True ﬂux: F? = 42

Frequentist: F? ' 53.9 ± 8.1

Bayesian: F? ' 47.1 ± 8.0

Reﬁning the Prior Based on the Observations:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 7 / 36 (1) (N?) p(F?) ∝ 1 × p(F? |F?) × ... × p(F? |F?) |{z} |{z} | {z } new prior initial prior cumulated data

Hierarchical model: consistently perform this process on all the data, at the same time.

Diﬀerence Between the Two Approaches: Using an Informative Prior

Accounting for Additional Information: Example: the star is in a globular cluster& you know the distance. Prior: cluster mass function& mass/luminosity relation ⇒ luminosity function.

True ﬂux: F? = 42

Frequentist: F? ' 53.9 ± 8.1

Bayesian: F? ' 47.1 ± 8.0

Reﬁning the Prior Based on the Observations: Cumulation of data: if you perform a series of observations in this cluster:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 7 / 36 (1) (N?) ∝ 1 × p(F? |F?) × ... × p(F? |F?) |{z} | {z } initial prior cumulated data

Hierarchical model: consistently perform this process on all the data, at the same time.

Diﬀerence Between the Two Approaches: Using an Informative Prior

Accounting for Additional Information: Example: the star is in a globular cluster& you know the distance. Prior: cluster mass function& mass/luminosity relation ⇒ luminosity function.

True ﬂux: F? = 42

Frequentist: F? ' 53.9 ± 8.1

Bayesian: F? ' 47.1 ± 8.0

Reﬁning the Prior Based on the Observations: Cumulation of data: if you perform a series of observations in this cluster:

p(F?) |{z} new prior

F. Galliano (AIM) Astromind 2019, CEA/Saclay 7 / 36 (1) (N?) p(F? |F?) × ... × p(F? |F?) | {z } cumulated data

Hierarchical model: consistently perform this process on all the data, at the same time.

Diﬀerence Between the Two Approaches: Using an Informative Prior

Accounting for Additional Information: Example: the star is in a globular cluster& you know the distance. Prior: cluster mass function& mass/luminosity relation ⇒ luminosity function.

True ﬂux: F? = 42

Frequentist: F? ' 53.9 ± 8.1

Bayesian: F? ' 47.1 ± 8.0

Reﬁning the Prior Based on the Observations: Cumulation of data: if you perform a series of observations in this cluster:

p(F?) ∝ 1 × |{z} |{z} new prior initial prior

F. Galliano (AIM) Astromind 2019, CEA/Saclay 7 / 36 Hierarchical model: consistently perform this process on all the data, at the same time.

Diﬀerence Between the Two Approaches: Using an Informative Prior

Accounting for Additional Information: Example: the star is in a globular cluster& you know the distance. Prior: cluster mass function& mass/luminosity relation ⇒ luminosity function.

True ﬂux: F? = 42

Frequentist: F? ' 53.9 ± 8.1

Bayesian: F? ' 47.1 ± 8.0

Reﬁning the Prior Based on the Observations: Cumulation of data: if you perform a series of observations in this cluster: (1) (N?) p(F?) ∝ 1 × p(F? |F?) × ... × p(F? |F?) |{z} |{z} | {z } new prior initial prior cumulated data

F. Galliano (AIM) Astromind 2019, CEA/Saclay 7 / 36 Diﬀerence Between the Two Approaches: Using an Informative Prior

Accounting for Additional Information: Example: the star is in a globular cluster& you know the distance. Prior: cluster mass function& mass/luminosity relation ⇒ luminosity function.

True ﬂux: F? = 42

Frequentist: F? ' 53.9 ± 8.1

Bayesian: F? ' 47.1 ± 8.0

Hierarchical model: consistently perform this process on all the data, at the same time.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 7 / 36 Noise: heavily-skewed split-normal distribution. Common sense: true flux< minimum measure: F? . 47.6. Frequentism: 95 % confidence interval: [47.5, 62.5] ⇒ inconsistent solution. Bayesianism: 95 % credible range: [34.6, 47.7] ⇒ consistent solution. This is an example of the Jeffreys-Lindley’s paradox.

⇒ question frequentist conﬁdence intervals& p-values. Bayesians address the question everyone is inter- ested in by using assumptions no-one believes, while Frequentists use impeccable logic to deal with an issue of no interest to anyone. (Lyons, 2013)

Same Problem with Asymmetric Noise:

(Example adapted from Jaynes, 1976)

Diﬃculty of Interpreting Frequentist Results: The frequentist interpretation leads to irrelevant interpretations

Diﬀerence Between the Two Approaches: Lindley’s Paradox

F. Galliano (AIM) Astromind 2019, CEA/Saclay 8 / 36 ⇒ question frequentist conﬁdence intervals& p-values. Bayesians address the question everyone is inter- ested in by using assumptions no-one believes, while Frequentists use impeccable logic to deal with an issue of no interest to anyone. (Lyons, 2013)

Noise: heavily-skewed split-normal distribution. Common sense: true flux< minimum measure: F? . 47.6. Frequentism: 95 % confidence interval: [47.5, 62.5] ⇒ inconsistent solution. Bayesianism: 95 % credible range: [34.6, 47.7] ⇒ consistent solution. This is an example of the Jeffreys-Lindley’s paradox. (Example adapted from Jaynes, 1976)

Diﬃculty of Interpreting Frequentist Results: The frequentist interpretation leads to irrelevant interpretations

Diﬀerence Between the Two Approaches: Lindley’s Paradox

Same Problem with Asymmetric Noise:

Diﬃculty of Interpreting Frequentist Results: The frequentist interpretation leads to irrelevant interpretations

Diﬀerence Between the Two Approaches: Lindley’s Paradox

Same Problem with Asymmetric Noise: True

. PDF/max .

(Example adapted from Jaynes, 1976)

Common sense: true flux< minimum measure: F? . 47.6. Frequentism: 95 % confidence interval: [47.5, 62.5] ⇒ inconsistent solution. Bayesianism: 95 % credible range: [34.6, 47.7] ⇒ consistent solution. This is an example of the Jeffreys-Lindley’s paradox.

Diﬃculty of Interpreting Frequentist Results: The frequentist interpretation leads to irrelevant interpretations

Diﬀerence Between the Two Approaches: Lindley’s Paradox

Same Problem with Asymmetric Noise: True Noise: heavily-skewed split-normal . distribution.

. PDF/max .

(Example adapted from Jaynes, 1976)

Diﬃculty of Interpreting Frequentist Results: The frequentist interpretation leads to irrelevant interpretations

Diﬀerence Between the Two Approaches: Lindley’s Paradox

Same Problem with Asymmetric Noise: True Noise: heavily-skewed split-normal . distribution.

. PDF/max .

(Example adapted from Jaynes, 1976)

Diﬃculty of Interpreting Frequentist Results: The frequentist interpretation leads to irrelevant interpretations

Diﬀerence Between the Two Approaches: Lindley’s Paradox

Same Problem with Asymmetric Noise: True Noise: heavily-skewed split-normal . distribution.

PDF/max Measures .

. PDF/max .

(Example adapted from Jaynes, 1976)

Diﬃculty of Interpreting Frequentist Results: The frequentist interpretation leads to irrelevant interpretations

Diﬀerence Between the Two Approaches: Lindley’s Paradox

Same Problem with Asymmetric Noise: True Noise: heavily-skewed split-normal . distribution.

PDF/max Measures .

. PDF/max .

(Example adapted from Jaynes, 1976)

Diﬃculty of Interpreting Frequentist Results: The frequentist interpretation leads to irrelevant interpretations

Diﬀerence Between the Two Approaches: Lindley’s Paradox

Same Problem with Asymmetric Noise: True Noise: heavily-skewed split-normal . distribution.

PDF/max Measures .

. PDF/max .

(Example adapted from Jaynes, 1976)

Frequentism: 95 % conﬁdence interval: [47.5, 62.5] ⇒ inconsistent solution. Bayesianism: 95 % credible range: [34.6, 47.7] ⇒ consistent solution. This is an example of the Jeﬀreys-Lindley’s paradox.

Diﬃculty of Interpreting Frequentist Results: The frequentist interpretation leads to irrelevant interpretations

Diﬀerence Between the Two Approaches: Lindley’s Paradox

Same Problem with Asymmetric Noise: True Noise: heavily-skewed split-normal . distribution. Common sense: true ﬂux< minimum measure: . F? . 47.6.

PDF/max Measures .

. PDF/max .

(Example adapted from Jaynes, 1976)

Diﬃculty of Interpreting Frequentist Results: The frequentist interpretation leads to irrelevant interpretations

Diﬀerence Between the Two Approaches: Lindley’s Paradox

Same Problem with Asymmetric Noise: True Noise: heavily-skewed split-normal . distribution. Common sense: true ﬂux< minimum measure: . F? . 47.6.

PDF/max Measures .

. PDF/max .

(Example adapted from Jaynes, 1976)

Diﬃculty of Interpreting Frequentist Results: The frequentist interpretation leads to irrelevant interpretations

Diﬀerence Between the Two Approaches: Lindley’s Paradox

Same Problem with Asymmetric Noise: True Noise: heavily-skewed split-normal . distribution. Common sense: true ﬂux< minimum measure: . F? . 47.6.

PDF/max Measures .

. PDF/max .

(Example adapted from Jaynes, 1976)

Diﬃculty of Interpreting Frequentist Results: The frequentist interpretation leads to irrelevant interpretations

Diﬀerence Between the Two Approaches: Lindley’s Paradox

Same Problem with Asymmetric Noise: True Noise: heavily-skewed split-normal . distribution. Common sense: true ﬂux< minimum measure: . F? . 47.6.

PDF/max Measures .

PDF/max Freq. .

(Example adapted from Jaynes, 1976)

Bayesianism: 95 % credible range: [34.6, 47.7] ⇒ consistent solution. This is an example of the Jeﬀreys-Lindley’s paradox.

Diﬃculty of Interpreting Frequentist Results: The frequentist interpretation leads to irrelevant interpretations

Diﬀerence Between the Two Approaches: Lindley’s Paradox

Same Problem with Asymmetric Noise: True Noise: heavily-skewed split-normal . distribution. Common sense: true ﬂux< minimum measure: . F? . 47.6. Frequentism: 95 % conﬁdence interval: [47.5, 62.5] PDF/max Measures ⇒ inconsistent solution. .

PDF/max Freq. .

(Example adapted from Jaynes, 1976)

Bayesianism: 95 % credible range: [34.6, 47.7] ⇒ consistent solution. This is an example of the Jeﬀreys-Lindley’s paradox.

Diﬃculty of Interpreting Frequentist Results: The frequentist interpretation leads to irrelevant interpretations

Diﬀerence Between the Two Approaches: Lindley’s Paradox

Bayes .

PDF/max Freq. .

(Example adapted from Jaynes, 1976)

This is an example of the Jeﬀreys-Lindley’s paradox.

Diﬃculty of Interpreting Frequentist Results: The frequentist interpretation leads to irrelevant interpretations

Diﬀerence Between the Two Approaches: Lindley’s Paradox

Bayes .

PDF/max Freq. .

(Example adapted from Jaynes, 1976)

Diﬃculty of Interpreting Frequentist Results: The frequentist interpretation leads to irrelevant interpretations

Diﬀerence Between the Two Approaches: Lindley’s Paradox

This is an example of the Jeﬀreys-Lindley’s paradox. Bayes .

PDF/max Freq. .

(Example adapted from Jaynes, 1976)

Diﬀerence Between the Two Approaches: Lindley’s Paradox

This is an example of the Jeﬀreys-Lindley’s paradox. Bayes . Diﬃculty of Interpreting Frequentist Results: The frequentist interpretation leads to irrelevant . interpretations PDF/max Freq. .

(Example adapted from Jaynes, 1976)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 8 / 36 Bayesians address the question everyone is inter- ested in by using assumptions no-one believes, while Frequentists use impeccable logic to deal with an issue of no interest to anyone. (Lyons, 2013)

Diﬀerence Between the Two Approaches: Lindley’s Paradox

(Example adapted from Jaynes, 1976)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 8 / 36 Diﬀerence Between the Two Approaches: Lindley’s Paradox

This is an example of the Jeffreys-Lindley’s paradox. Bayes . Difficulty of Interpreting Frequentist Results: The frequentist interpretation leads to irrelevant . interpretations ⇒ question frequentist confidence PDF/max Freq. intervals& p-values. . Bayesians address the question everyone is inter- ested in by using assumptions no-one believes, while Frequentists use impeccable logic to deal with an issue of no interest to anyone. (Lyons, 2013) (Example adapted from Jaynes, 1976)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 8 / 36 ⇒ “post p<0.05 era”.

p(H |data) p(data|H ) p(H ) H0: the null 1 = 1 × 1 hypothesis; p(H0|data) p(data|H0) p(H0) | {z } | {z } | {z } posterior odds Bayes factor prior odds H1: complement. Bayes factor Strength of evidence Commonly: 1 to 3.2 Barely worth mentioning p < 0.05 3.2 to 10 Substantial ⇒ reject H . 10 to 100 Strong 0 > 100 Decisive (Jeﬀreys, 1961)

2011: concept of p-hacking (Simmons et al., 2011). 2015: Basic & Applied Social Psychology “would no longer publish papers containing p-values because they were too often used to support lower-quality research”. 2016: statement of the American Statistical Association: “widespread use of ’statistical significance’ (generally interpreted as ’p<0.05’) as a license for making a claim of a scientific finding (or implied truth) leads to considerable distortion of the scientific process” 2018: Political Analysis “will no longer be reporting p-values”.

Bayesian Hypothesis Testing: Frequentist Hypothesis Testing:

Recent Controversy About the Interpretation & the Signiﬁcance of p-Values:

Hypothesis Testing: the Limitations of Frequentist p-Values

F. Galliano (AIM) Astromind 2019, CEA/Saclay 9 / 36 ⇒ “post p<0.05 era”.

H0: the null hypothesis;

H1: complement.

Commonly: p < 0.05 ⇒ reject H0.

Frequentist Hypothesis Testing:

p(H |data) p(data|H ) p(H ) 1 = 1 × 1 p(H0|data) p(data|H0) p(H0) | {z } | {z } | {z } posterior odds Bayes factor prior odds Bayes factor Strength of evidence 1 to 3.2 Barely worth mentioning 3.2 to 10 Substantial 10 to 100 Strong > 100 Decisive (Jeﬀreys, 1961)

Recent Controversy About the Interpretation & the Signiﬁcance of p-Values:

Hypothesis Testing: the Limitations of Frequentist p-Values

Bayesian Hypothesis Testing:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 9 / 36 ⇒ “post p<0.05 era”.

H0: the null hypothesis;

H1: complement.

Commonly: p < 0.05 ⇒ reject H0.

Frequentist Hypothesis Testing:

p(data|H ) p(H ) 1 × 1 p(data|H0) p(H0) | {z } | {z } Bayes factor prior odds Bayes factor Strength of evidence 1 to 3.2 Barely worth mentioning 3.2 to 10 Substantial 10 to 100 Strong > 100 Decisive (Jeﬀreys, 1961)

Recent Controversy About the Interpretation & the Signiﬁcance of p-Values:

Hypothesis Testing: the Limitations of Frequentist p-Values

Bayesian Hypothesis Testing:

p(H |data) 1 = p(H0|data) | {z } posterior odds

F. Galliano (AIM) Astromind 2019, CEA/Saclay 9 / 36 ⇒ “post p<0.05 era”.

H0: the null hypothesis;

H1: complement.

Commonly: p < 0.05 ⇒ reject H0.

Frequentist Hypothesis Testing:

p(data|H ) 1 × p(data|H0) | {z } Bayes factor Bayes factor Strength of evidence 1 to 3.2 Barely worth mentioning 3.2 to 10 Substantial 10 to 100 Strong > 100 Decisive (Jeﬀreys, 1961)

Recent Controversy About the Interpretation & the Signiﬁcance of p-Values:

Hypothesis Testing: the Limitations of Frequentist p-Values

Bayesian Hypothesis Testing:

p(H |data) p(H ) 1 = 1 p(H0|data) p(H0) | {z } | {z } posterior odds prior odds

F. Galliano (AIM) Astromind 2019, CEA/Saclay 9 / 36 ⇒ “post p<0.05 era”.

H0: the null hypothesis;

H1: complement.

Commonly: p < 0.05 ⇒ reject H0.

Frequentist Hypothesis Testing:

Bayes factor Strength of evidence 1 to 3.2 Barely worth mentioning 3.2 to 10 Substantial 10 to 100 Strong > 100 Decisive (Jeﬀreys, 1961)

Recent Controversy About the Interpretation & the Signiﬁcance of p-Values:

Hypothesis Testing: the Limitations of Frequentist p-Values

Bayesian Hypothesis Testing:

p(H |data) p(data|H ) p(H ) 1 = 1 × 1 p(H0|data) p(data|H0) p(H0) | {z } | {z } | {z } posterior odds Bayes factor prior odds

F. Galliano (AIM) Astromind 2019, CEA/Saclay 9 / 36 ⇒ “post p<0.05 era”.

H0: the null hypothesis;

H1: complement.

Commonly: p < 0.05 ⇒ reject H0.

Frequentist Hypothesis Testing:

Recent Controversy About the Interpretation & the Signiﬁcance of p-Values:

Hypothesis Testing: the Limitations of Frequentist p-Values

Bayesian Hypothesis Testing:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 9 / 36 ⇒ “post p<0.05 era”.

H0: the null hypothesis;

H1: complement.

Commonly: p < 0.05 ⇒ reject H0.

Recent Controversy About the Interpretation & the Signiﬁcance of p-Values:

Hypothesis Testing: the Limitations of Frequentist p-Values

Bayesian Hypothesis Testing: Frequentist Hypothesis Testing:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 9 / 36 ⇒ “post p<0.05 era”.

H0: the null hypothesis;

H1: complement.

Commonly: p < 0.05 ⇒ reject H0.

Recent Controversy About the Interpretation & the Signiﬁcance of p-Values:

Hypothesis Testing: the Limitations of Frequentist p-Values

Bayesian Hypothesis Testing: Frequentist Hypothesis Testing:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 9 / 36 ⇒ “post p<0.05 era”.

H1: complement.

Commonly: p < 0.05 ⇒ reject H0.

Recent Controversy About the Interpretation & the Signiﬁcance of p-Values:

Hypothesis Testing: the Limitations of Frequentist p-Values

Bayesian Hypothesis Testing: Frequentist Hypothesis Testing:

p(H |data) p(data|H ) p(H ) H0: the null 1 = 1 × 1 hypothesis; p(H0|data) p(data|H0) p(H0) | {z } | {z } | {z } posterior odds Bayes factor prior odds Bayes factor Strength of evidence 1 to 3.2 Barely worth mentioning 3.2 to 10 Substantial 10 to 100 Strong > 100 Decisive (Jeﬀreys, 1961)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 9 / 36 ⇒ “post p<0.05 era”.

Commonly: p < 0.05 ⇒ reject H0.

Recent Controversy About the Interpretation & the Signiﬁcance of p-Values:

Hypothesis Testing: the Limitations of Frequentist p-Values

Bayesian Hypothesis Testing: Frequentist Hypothesis Testing:

p(H |data) p(data|H ) p(H ) H0: the null 1 = 1 × 1 hypothesis; p(H0|data) p(data|H0) p(H0) | {z } | {z } | {z } posterior odds Bayes factor prior odds H1: complement. Bayes factor Strength of evidence 1 to 3.2 Barely worth mentioning 3.2 to 10 Substantial 10 to 100 Strong > 100 Decisive (Jeﬀreys, 1961)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 9 / 36 ⇒ “post p<0.05 era”.

Commonly: p < 0.05 ⇒ reject H0.

Recent Controversy About the Interpretation & the Signiﬁcance of p-Values:

Hypothesis Testing: the Limitations of Frequentist p-Values

Bayesian Hypothesis Testing: Frequentist Hypothesis Testing:

p(H |data) p(data|H ) p(H ) H0: the null 1 = 1 × 1 hypothesis; p(H0|data) p(data|H0) p(H0) | {z } | {z } | {z } posterior odds Bayes factor prior odds H1: complement. Bayes factor Strength of evidence 1 to 3.2 Barely worth mentioning 3.2 to 10 Substantial 10 to 100 Strong > 100 Decisive (Jeﬀreys, 1961)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 9 / 36 ⇒ “post p<0.05 era”.

Recent Controversy About the Interpretation & the Signiﬁcance of p-Values:

Hypothesis Testing: the Limitations of Frequentist p-Values

Bayesian Hypothesis Testing: Frequentist Hypothesis Testing:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 9 / 36 ⇒ “post p<0.05 era”.

Hypothesis Testing: the Limitations of Frequentist p-Values

Bayesian Hypothesis Testing: Frequentist Hypothesis Testing:

Recent Controversy About the Interpretation & the Signiﬁcance of p-Values:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 9 / 36 ⇒ “post p<0.05 era”.

2015: Basic & Applied Social Psychology “would no longer publish papers containing p-values because they were too often used to support lower-quality research”. 2016: statement of the American Statistical Association: “widespread use of ’statistical significance’ (generally interpreted as ’p<0.05’) as a license for making a claim of a scientific finding (or implied truth) leads to considerable distortion of the scientific process” 2018: Political Analysis “will no longer be reporting p-values”.

Hypothesis Testing: the Limitations of Frequentist p-Values

Bayesian Hypothesis Testing: Frequentist Hypothesis Testing:

Recent Controversy About the Interpretation & the Signiﬁcance of p-Values: 2011: concept of p-hacking (Simmons et al., 2011).

F. Galliano (AIM) Astromind 2019, CEA/Saclay 9 / 36 ⇒ “post p<0.05 era”.

2016: statement of the American Statistical Association: “widespread use of ’statistical significance’ (generally interpreted as ’p<0.05’) as a license for making a claim of a scientific finding (or implied truth) leads to considerable distortion of the scientific process” 2018: Political Analysis “will no longer be reporting p-values”.

Hypothesis Testing: the Limitations of Frequentist p-Values

Bayesian Hypothesis Testing: Frequentist Hypothesis Testing:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 9 / 36 ⇒ “post p<0.05 era”. 2018: Political Analysis “will no longer be reporting p-values”.

Hypothesis Testing: the Limitations of Frequentist p-Values

Bayesian Hypothesis Testing: Frequentist Hypothesis Testing:

Recent Controversy About the Interpretation & the Significance of p-Values: 2011: concept of p-hacking (Simmons et al., 2011). 2015: Basic & Applied Social Psychology “would no longer publish papers containing p-values because they were too often used to support lower-quality research”. 2016: statement of the American Statistical Association: “widespread use of ’statistical significance’ (generally interpreted as ’p<0.05’) as a license for making a claim of a scientific finding (or implied truth) leads to considerable distortion of the scientific process”

F. Galliano (AIM) Astromind 2019, CEA/Saclay 9 / 36 2018: Political Analysis “will no longer be reporting p-values”.

Hypothesis Testing: the Limitations of Frequentist p-Values

Bayesian Hypothesis Testing: Frequentist Hypothesis Testing:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 9 / 36 Hypothesis Testing: the Limitations of Frequentist p-Values

Bayesian Hypothesis Testing: Frequentist Hypothesis Testing:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 9 / 36 Probability of an event, knowing it happened n consecutive times = (n + 1)/(n + 2). The Sun is 5 000 years ⇒ n = 1 826 213 (Bible). p(nova) = 1 − p(sunrise) = 1/1 826 215.

p(2 × 6) = 1/36 p(2 × 6|nova) = 1 p(nova) = 1/1 826 215 ⇒ p(nova|2 × 6) ' 2 × 10−5

The Bayesian Point of View:

p(2 × 6|nova) × p(nova) p(nova|2 × 6) = p(2 × 6)

Laplace’s Law of Succession:

(Essai philosophique sur les probabilités, 1814)

The Bayesian Solution:

Reductio Ad Absurdum (xkcd)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 10 / 36 Probability of an event, knowing it happened n consecutive times = (n + 1)/(n + 2). The Sun is 5 000 years ⇒ n = 1 826 213 (Bible). p(nova) = 1 − p(sunrise) = 1/1 826 215.

p(2 × 6) = 1/36 p(2 × 6|nova) = 1 p(nova) = 1/1 826 215 ⇒ p(nova|2 × 6) ' 2 × 10−5

The Bayesian Point of View:

p(2 × 6|nova) × p(nova) p(nova|2 × 6) = p(2 × 6)

Laplace’s Law of Succession:

(Essai philosophique sur les probabilités, 1814)

The Bayesian Solution:

Reductio Ad Absurdum (xkcd)

p(2 × 6) = 1/36 p(2 × 6|nova) = 1 p(nova) = 1/1 826 215 ⇒ p(nova|2 × 6) ' 2 × 10−5

The Bayesian Point of View:

p(2 × 6|nova) × p(nova) p(nova|2 × 6) = p(2 × 6)

Laplace’s Law of Succession:

(Essai philosophique sur les probabilités, 1814)

The Bayesian Solution:

Reductio Ad Absurdum (xkcd)

p(2 × 6) = 1/36 p(2 × 6|nova) = 1 p(nova) = 1/1 826 215 ⇒ p(nova|2 × 6) ' 2 × 10−5

The Bayesian Point of View:

p(2 × 6|nova) × p(nova) p(nova|2 × 6) = p(2 × 6)

Laplace’s Law of Succession:

(Essai philosophique sur les probabilités, 1814)

The Bayesian Solution:

Reductio Ad Absurdum (xkcd)

p(2 × 6) = 1/36 p(2 × 6|nova) = 1 p(nova) = 1/1 826 215 ⇒ p(nova|2 × 6) ' 2 × 10−5

Laplace’s Law of Succession:

(Essai philosophique sur les probabilités, 1814)

The Bayesian Solution:

Reductio Ad Absurdum (xkcd)

The Bayesian Point of View:

p(2 × 6|nova) × p(nova) p(nova|2 × 6) = p(2 × 6)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 10 / 36 p(2 × 6) = 1/36 p(2 × 6|nova) = 1 p(nova) = 1/1 826 215 ⇒ p(nova|2 × 6) ' 2 × 10−5

Probability of an event, knowing it happened n consecutive times = (n + 1)/(n + 2). The Sun is 5 000 years ⇒ n = 1 826 213 (Bible). p(nova) = 1 − p(sunrise) = 1/1 826 215.

The Bayesian Solution:

Reductio Ad Absurdum (xkcd)

The Bayesian Point of View:

p(2 × 6|nova) × p(nova) p(nova|2 × 6) = p(2 × 6)

Laplace’s Law of Succession:

(Essai philosophique sur les probabilités, 1814)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 10 / 36 p(2 × 6) = 1/36 p(2 × 6|nova) = 1 p(nova) = 1/1 826 215 ⇒ p(nova|2 × 6) ' 2 × 10−5

The Sun is 5 000 years ⇒ n = 1 826 213 (Bible). p(nova) = 1 − p(sunrise) = 1/1 826 215.

The Bayesian Solution:

Reductio Ad Absurdum (xkcd)

The Bayesian Point of View:

p(2 × 6|nova) × p(nova) p(nova|2 × 6) = p(2 × 6)

Laplace’s Law of Succession: Probability of an event, knowing it happened n consecutive times = (n + 1)/(n + 2).

(Essai philosophique sur les probabilités, 1814)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 10 / 36 p(2 × 6) = 1/36 p(2 × 6|nova) = 1 p(nova) = 1/1 826 215 ⇒ p(nova|2 × 6) ' 2 × 10−5

p(nova) = 1 − p(sunrise) = 1/1 826 215.

The Bayesian Solution:

Reductio Ad Absurdum (xkcd)

The Bayesian Point of View:

p(2 × 6|nova) × p(nova) p(nova|2 × 6) = p(2 × 6)

Laplace’s Law of Succession: Probability of an event, knowing it happened n consecutive times = (n + 1)/(n + 2). The Sun is 5 000 years ⇒ n = 1 826 213 (Bible).

(Essai philosophique sur les probabilités, 1814)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 10 / 36 p(2 × 6) = 1/36 p(2 × 6|nova) = 1 p(nova) = 1/1 826 215 ⇒ p(nova|2 × 6) ' 2 × 10−5

The Bayesian Solution:

Reductio Ad Absurdum (xkcd)

The Bayesian Point of View:

p(2 × 6|nova) × p(nova) p(nova|2 × 6) = p(2 × 6)

Laplace’s Law of Succession: Probability of an event, knowing it happened n consecutive times = (n + 1)/(n + 2). The Sun is 5 000 years ⇒ n = 1 826 213 (Bible). p(nova) = 1 − p(sunrise) = 1/1 826 215.

(Essai philosophique sur les probabilités, 1814)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 10 / 36 p(2 × 6) = 1/36 p(2 × 6|nova) = 1 p(nova) = 1/1 826 215 ⇒ p(nova|2 × 6) ' 2 × 10−5

Reductio Ad Absurdum (xkcd)

The Bayesian Point of View:

p(2 × 6|nova) × p(nova) p(nova|2 × 6) = p(2 × 6)

(Essai philosophique sur les probabilités, 1814)

The Bayesian Solution:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 10 / 36 p(2 × 6|nova) = 1 p(nova) = 1/1 826 215 ⇒ p(nova|2 × 6) ' 2 × 10−5

Reductio Ad Absurdum (xkcd)

The Bayesian Point of View:

p(2 × 6|nova) × p(nova) p(nova|2 × 6) = p(2 × 6)

(Essai philosophique sur les probabilités, 1814)

The Bayesian Solution:

p(2 × 6) = 1/36

F. Galliano (AIM) Astromind 2019, CEA/Saclay 10 / 36 p(nova) = 1/1 826 215 ⇒ p(nova|2 × 6) ' 2 × 10−5

Reductio Ad Absurdum (xkcd)

The Bayesian Point of View:

p(2 × 6|nova) × p(nova) p(nova|2 × 6) = p(2 × 6)

(Essai philosophique sur les probabilités, 1814)

The Bayesian Solution:

p(2 × 6) = 1/36 p(2 × 6|nova) = 1

F. Galliano (AIM) Astromind 2019, CEA/Saclay 10 / 36 ⇒ p(nova|2 × 6) ' 2 × 10−5

Reductio Ad Absurdum (xkcd)

The Bayesian Point of View:

p(2 × 6|nova) × p(nova) p(nova|2 × 6) = p(2 × 6)

(Essai philosophique sur les probabilités, 1814)

The Bayesian Solution:

p(2 × 6) = 1/36 p(2 × 6|nova) = 1 p(nova) = 1/1 826 215

F. Galliano (AIM) Astromind 2019, CEA/Saclay 10 / 36 Reductio Ad Absurdum (xkcd)

The Bayesian Point of View:

p(2 × 6|nova) × p(nova) p(nova|2 × 6) = p(2 × 6)

(Essai philosophique sur les probabilités, 1814)

The Bayesian Solution:

p(2 × 6) = 1/36 p(2 × 6|nova) = 1 p(nova) = 1/1 826 215 ⇒ p(nova|2 × 6) ' 2 × 10−5

F. Galliano (AIM) Astromind 2019, CEA/Saclay 10 / 36 Prior Depends on Dust Model Parameters: Prior Also Includes Ancillary Data:

Intuitive approach: p(Mdust). Holistic approach: p(Mdust, Mgas). (Galliano, 2018)

⇒ partition of knowledge is statistically inadmissible (Bayesian take on Stein’s paradox).

The Bayesian Approach is Holistic: Stein’s Paradox

F. Galliano (AIM) Astromind 2019, CEA/Saclay 11 / 36 Prior Also Includes Ancillary Data:

Holistic approach: p(Mdust, Mgas). (Galliano, 2018)

⇒ partition of knowledge is statistically inadmissible (Bayesian take on Stein’s paradox).

The Bayesian Approach is Holistic: Stein’s Paradox

Prior Depends on Dust Model Parameters:

Intuitive approach: p(Mdust).

F. Galliano (AIM) Astromind 2019, CEA/Saclay 11 / 36 Prior Also Includes Ancillary Data:

Holistic approach: p(Mdust, Mgas).

⇒ partition of knowledge is statistically inadmissible (Bayesian take on Stein’s paradox).

The Bayesian Approach is Holistic: Stein’s Paradox

Prior Depends on Dust Model Parameters:

Intuitive approach: p(Mdust).

(Galliano, 2018)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 11 / 36 Prior Also Includes Ancillary Data:

Holistic approach: p(Mdust, Mgas).

⇒ partition of knowledge is statistically inadmissible (Bayesian take on Stein’s paradox).

The Bayesian Approach is Holistic: Stein’s Paradox

Prior Depends on Dust Model Parameters:

Intuitive approach: p(Mdust).

(Galliano, 2018)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 11 / 36 ⇒ partition of knowledge is statistically inadmissible (Bayesian take on Stein’s paradox).

The Bayesian Approach is Holistic: Stein’s Paradox

Prior Depends on Dust Model Parameters: Prior Also Includes Ancillary Data:

Intuitive approach: p(Mdust). Holistic approach: p(Mdust, Mgas).

(Galliano, 2018)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 11 / 36 ⇒ partition of knowledge is statistically inadmissible (Bayesian take on Stein’s paradox).

The Bayesian Approach is Holistic: Stein’s Paradox

Prior Depends on Dust Model Parameters: Prior Also Includes Ancillary Data:

Intuitive approach: p(Mdust). Holistic approach: p(Mdust, Mgas).

(Galliano, 2018)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 11 / 36 ⇒ partition of knowledge is statistically inadmissible (Bayesian take on Stein’s paradox).

The Bayesian Approach is Holistic: Stein’s Paradox

Prior Depends on Dust Model Parameters: Prior Also Includes Ancillary Data:

Intuitive approach: p(Mdust). Holistic approach: p(Mdust, Mgas).

(Galliano, 2018)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 11 / 36 The Bayesian Approach is Holistic: Stein’s Paradox

Prior Depends on Dust Model Parameters: Prior Also Includes Ancillary Data:

Intuitive approach: p(Mdust). Holistic approach: p(Mdust, Mgas).

(Galliano, 2018) ⇒ partition of knowledge is statistically inadmissible (Bayesian take on Stein’s paradox). F. Galliano (AIM) Astromind 2019, CEA/Saclay 11 / 36 CON choice of prior is arbitrary. PRO likelihood is not arbitrary.

PRO the posterior makes sense CON samples non-observed data& (conditional on the data)& arbitrary choice of estimator, is easy to interpret. loss functions or p-value. PRO probabilistic logic CON boolean logic ⇒ a ⇒ continuum between proposition is either true or skepticism& conﬁdence. false. CON heavy computation. PRO fast computation.

PRO works well with small CON does not work well with small samples, even with 0 data. . . samples. PRO master equation ⇒ easier to CON diﬃcult to teach (collection teach. of ad hoc cooking recipes). PRO holistic& ﬂexible: can CON strict: can account only for account for all data& data related to a particular theories. experiment. PRO conservative. CON can give ridiculous answers.

The Bayesian Approach The Frequentist Approach

(cf. reviews by e.g. Jaynes, 1976; Loredo, 1990; Lindley, 1999; Wagenmacker, 2008; Lyons, 2013)

Pros & Cons of the Two Approaches

F. Galliano (AIM) Astromind 2019, CEA/Saclay 12 / 36 CON choice of prior is arbitrary. PRO likelihood is not arbitrary.

Pros & Cons of the Two Approaches

The Bayesian Approach The Frequentist Approach

(cf. reviews by e.g. Jaynes, 1976; Loredo, 1990; Lindley, 1999; Wagenmacker, 2008; Lyons, 2013)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 12 / 36 PRO likelihood is not arbitrary.

Pros & Cons of the Two Approaches

The Bayesian Approach The Frequentist Approach

CON choice of prior is arbitrary.

(cf. reviews by e.g. Jaynes, 1976; Loredo, 1990; Lindley, 1999; Wagenmacker, 2008; Lyons, 2013)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 12 / 36 PRO the posterior makes sense CON samples non-observed data& (conditional on the data)& arbitrary choice of estimator, is easy to interpret. loss functions or p-value. PRO probabilistic logic CON boolean logic ⇒ a ⇒ continuum between proposition is either true or skepticism& conﬁdence. false. CON heavy computation. PRO fast computation.

Pros & Cons of the Two Approaches

The Bayesian Approach The Frequentist Approach

CON choice of prior is arbitrary. PRO likelihood is not arbitrary.

(cf. reviews by e.g. Jaynes, 1976; Loredo, 1990; Lindley, 1999; Wagenmacker, 2008; Lyons, 2013)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 12 / 36 CON samples non-observed data& arbitrary choice of estimator, loss functions or p-value. PRO probabilistic logic CON boolean logic ⇒ a ⇒ continuum between proposition is either true or skepticism& conﬁdence. false. CON heavy computation. PRO fast computation.

Pros & Cons of the Two Approaches

The Bayesian Approach The Frequentist Approach

CON choice of prior is arbitrary. PRO likelihood is not arbitrary.

PRO the posterior makes sense (conditional on the data)& is easy to interpret.

(cf. reviews by e.g. Jaynes, 1976; Loredo, 1990; Lindley, 1999; Wagenmacker, 2008; Lyons, 2013)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 12 / 36 PRO probabilistic logic CON boolean logic ⇒ a ⇒ continuum between proposition is either true or skepticism& conﬁdence. false. CON heavy computation. PRO fast computation.

Pros & Cons of the Two Approaches

The Bayesian Approach The Frequentist Approach

CON choice of prior is arbitrary. PRO likelihood is not arbitrary.

PRO the posterior makes sense CON samples non-observed data& (conditional on the data)& arbitrary choice of estimator, is easy to interpret. loss functions or p-value.

(cf. reviews by e.g. Jaynes, 1976; Loredo, 1990; Lindley, 1999; Wagenmacker, 2008; Lyons, 2013)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 12 / 36 CON boolean logic ⇒ a proposition is either true or false. CON heavy computation. PRO fast computation.

Pros & Cons of the Two Approaches

The Bayesian Approach The Frequentist Approach

CON choice of prior is arbitrary. PRO likelihood is not arbitrary.

(cf. reviews by e.g. Jaynes, 1976; Loredo, 1990; Lindley, 1999; Wagenmacker, 2008; Lyons, 2013)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 12 / 36 CON heavy computation. PRO fast computation.

Pros & Cons of the Two Approaches

The Bayesian Approach The Frequentist Approach

CON choice of prior is arbitrary. PRO likelihood is not arbitrary.

(cf. reviews by e.g. Jaynes, 1976; Loredo, 1990; Lindley, 1999; Wagenmacker, 2008; Lyons, 2013)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 12 / 36 PRO fast computation.

Pros & Cons of the Two Approaches

The Bayesian Approach The Frequentist Approach

CON choice of prior is arbitrary. PRO likelihood is not arbitrary.

(cf. reviews by e.g. Jaynes, 1976; Loredo, 1990; Lindley, 1999; Wagenmacker, 2008; Lyons, 2013)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 12 / 36 PRO works well with small CON does not work well with small samples, even with 0 data. . . samples. PRO master equation ⇒ easier to CON diﬃcult to teach (collection teach. of ad hoc cooking recipes). PRO holistic& ﬂexible: can CON strict: can account only for account for all data& data related to a particular theories. experiment. PRO conservative. CON can give ridiculous answers.

Pros & Cons of the Two Approaches

The Bayesian Approach The Frequentist Approach

CON choice of prior is arbitrary. PRO likelihood is not arbitrary.

(cf. reviews by e.g. Jaynes, 1976; Loredo, 1990; Lindley, 1999; Wagenmacker, 2008; Lyons, 2013)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 12 / 36 CON does not work well with small samples.

PRO master equation ⇒ easier to CON diﬃcult to teach (collection teach. of ad hoc cooking recipes). PRO holistic& ﬂexible: can CON strict: can account only for account for all data& data related to a particular theories. experiment. PRO conservative. CON can give ridiculous answers.

Pros & Cons of the Two Approaches

The Bayesian Approach The Frequentist Approach

CON choice of prior is arbitrary. PRO likelihood is not arbitrary.

PRO works well with small samples, even with 0 data. . .

(cf. reviews by e.g. Jaynes, 1976; Loredo, 1990; Lindley, 1999; Wagenmacker, 2008; Lyons, 2013)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 12 / 36 PRO master equation ⇒ easier to CON diﬃcult to teach (collection teach. of ad hoc cooking recipes). PRO holistic& ﬂexible: can CON strict: can account only for account for all data& data related to a particular theories. experiment. PRO conservative. CON can give ridiculous answers.

Pros & Cons of the Two Approaches

The Bayesian Approach The Frequentist Approach

CON choice of prior is arbitrary. PRO likelihood is not arbitrary.

PRO works well with small CON does not work well with small samples, even with 0 data. . . samples.

(cf. reviews by e.g. Jaynes, 1976; Loredo, 1990; Lindley, 1999; Wagenmacker, 2008; Lyons, 2013)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 12 / 36 CON diﬃcult to teach (collection of ad hoc cooking recipes). PRO holistic& ﬂexible: can CON strict: can account only for account for all data& data related to a particular theories. experiment. PRO conservative. CON can give ridiculous answers.

Pros & Cons of the Two Approaches

The Bayesian Approach The Frequentist Approach

CON choice of prior is arbitrary. PRO likelihood is not arbitrary.

PRO works well with small CON does not work well with small samples, even with 0 data. . . samples. PRO master equation ⇒ easier to teach.

(cf. reviews by e.g. Jaynes, 1976; Loredo, 1990; Lindley, 1999; Wagenmacker, 2008; Lyons, 2013)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 12 / 36 PRO holistic& ﬂexible: can CON strict: can account only for account for all data& data related to a particular theories. experiment. PRO conservative. CON can give ridiculous answers.

Pros & Cons of the Two Approaches

The Bayesian Approach The Frequentist Approach

CON choice of prior is arbitrary. PRO likelihood is not arbitrary.

(cf. reviews by e.g. Jaynes, 1976; Loredo, 1990; Lindley, 1999; Wagenmacker, 2008; Lyons, 2013)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 12 / 36 CON strict: can account only for data related to a particular experiment. PRO conservative. CON can give ridiculous answers.

Pros & Cons of the Two Approaches

The Bayesian Approach The Frequentist Approach

CON choice of prior is arbitrary. PRO likelihood is not arbitrary.

(cf. reviews by e.g. Jaynes, 1976; Loredo, 1990; Lindley, 1999; Wagenmacker, 2008; Lyons, 2013)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 12 / 36 PRO conservative. CON can give ridiculous answers.

Pros & Cons of the Two Approaches

The Bayesian Approach The Frequentist Approach

CON choice of prior is arbitrary. PRO likelihood is not arbitrary.

(cf. reviews by e.g. Jaynes, 1976; Loredo, 1990; Lindley, 1999; Wagenmacker, 2008; Lyons, 2013)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 12 / 36 CON can give ridiculous answers.

Pros & Cons of the Two Approaches

The Bayesian Approach The Frequentist Approach

CON choice of prior is arbitrary. PRO likelihood is not arbitrary.

(cf. reviews by e.g. Jaynes, 1976; Loredo, 1990; Lindley, 1999; Wagenmacker, 2008; Lyons, 2013)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 12 / 36 Pros & Cons of the Two Approaches

The Bayesian Approach The Frequentist Approach

CON choice of prior is arbitrary. PRO likelihood is not arbitrary.

(cf. reviews by e.g. Jaynes, 1976; Loredo, 1990; Lindley, 1999; Wagenmacker, 2008; Lyons, 2013)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 12 / 36 Outline of the Talk

1 BAYESIANS VS FREQUENTISTS Epistemological Principles & Comparison Demonstration on a Simple Example Limitations of the Frequentist Approach

2 BAYES’ RULE THROUGH HISTORY Early Development The Frequentist Winter The Bayesian Renaissance

3 IMPLICATIONS FOR THE SCIENTIFIC METHOD Karl Popper’s Logic of Scientiﬁc Discovery Bayesian Epistemology How Researchers Actually Work

4 SUMMARY & CONCLUSION

F. Galliano (AIM) Astromind 2019, CEA/Saclay 13 / 36 Sharon Bertsch McGRAYNE (1942–)

(Published in 2011)

A Comprehensive Historical Perspective: S. McGrayne’s Book

F. Galliano (AIM) Astromind 2019, CEA/Saclay 14 / 36 A Comprehensive Historical Perspective: S. McGrayne’s Book

Sharon Bertsch McGRAYNE (1942–)

(Published in 2011)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 14 / 36 Presbyterian reverend from a nonconformist family. Read: (1) David HUME (philosophy); (2) Isaac NEWTON (physics); and (3) Abraham DE MOIVRE (proabilities). Published, during his lifetime: (1) a treaty of theology; and (2) a treaty of mathematics.

Thought Experiment: between 1746 and 1749, he formulates his rule to infer the position of a ball on a pool table. Prior: position relative to another ball. Probability of an event: “the ratio between the value at which an expectation depending on the happening of the event ought to be computed, and the value of the thing expected upon its happening” (An Essay towards solving a Problem in the Doctrine of Chances, 1764). Publication: in 1764, by Richard PRICE: his formula gives the probability of causes ⇒ can be applied to prove God’s existence.

Intellectual Interests:

Thomas BAYES ('1701–1761)

The Discovery of Bayes’ Rule:

First Formulation by Bayes

F. Galliano (AIM) Astromind 2019, CEA/Saclay 15 / 36 Presbyterian reverend from a nonconformist family. Read: (1) David HUME (philosophy); (2) Isaac NEWTON (physics); and (3) Abraham DE MOIVRE (proabilities). Published, during his lifetime: (1) a treaty of theology; and (2) a treaty of mathematics.

Intellectual Interests:

The Discovery of Bayes’ Rule:

First Formulation by Bayes

Thomas BAYES ('1701–1761)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 15 / 36 Thought Experiment: between 1746 and 1749, he formulates his rule to infer the position of a ball on a pool table. Prior: position relative to another ball. Probability of an event: “the ratio between the value at which an expectation depending on the happening of the event ought to be computed, and the value of the thing expected upon its happening” (An Essay towards solving a Problem in the Doctrine of Chances, 1764). Publication: in 1764, by Richard PRICE: his formula gives the probability of causes ⇒ can be applied to prove God’s existence.

Presbyterian reverend from a nonconformist family. Read: (1) David HUME (philosophy); (2) Isaac NEWTON (physics); and (3) Abraham DE MOIVRE (proabilities). Published, during his lifetime: (1) a treaty of theology; and (2) a treaty of mathematics.

The Discovery of Bayes’ Rule:

First Formulation by Bayes

Intellectual Interests:

Thomas BAYES ('1701–1761)

Read: (1) David HUME (philosophy); (2) Isaac NEWTON (physics); and (3) Abraham DE MOIVRE (proabilities). Published, during his lifetime: (1) a treaty of theology; and (2) a treaty of mathematics.

The Discovery of Bayes’ Rule:

First Formulation by Bayes

Intellectual Interests: Presbyterian reverend from a nonconformist family. Thomas BAYES ('1701–1761)

Published, during his lifetime: (1) a treaty of theology; and (2) a treaty of mathematics.

The Discovery of Bayes’ Rule:

First Formulation by Bayes

The Discovery of Bayes’ Rule:

First Formulation by Bayes

Intellectual Interests: Presbyterian reverend from a nonconformist family. Thomas BAYES Read: (1) David HUME (philosophy); ('1701–1761) (2) Isaac NEWTON (physics); and (3) Abraham DE MOIVRE (proabilities). Published, during his lifetime: (1) a treaty of theology; and (2) a treaty of mathematics.

First Formulation by Bayes

The Discovery of Bayes’ Rule:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 15 / 36 Probability of an event: “the ratio between the value at which an expectation depending on the happening of the event ought to be computed, and the value of the thing expected upon its happening” (An Essay towards solving a Problem in the Doctrine of Chances, 1764). Publication: in 1764, by Richard PRICE: his formula gives the probability of causes ⇒ can be applied to prove God’s existence.

First Formulation by Bayes

The Discovery of Bayes’ Rule: Thought Experiment: between 1746 and 1749, he formulates his rule to infer the position of a ball on a pool table. Prior: position relative to another ball.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 15 / 36 Publication: in 1764, by Richard PRICE: his formula gives the probability of causes ⇒ can be applied to prove God’s existence.

First Formulation by Bayes

The Discovery of Bayes’ Rule: Thought Experiment: between 1746 and 1749, he formulates his rule to infer the position of a ball on a pool table. Prior: position relative to another ball. Probability of an event: “the ratio between the value at which an expectation depending on the happening of the event ought to be computed, and the value of the thing expected upon its happening” (An Essay towards solving a Problem in the Doctrine of Chances, 1764).

F. Galliano (AIM) Astromind 2019, CEA/Saclay 15 / 36 First Formulation by Bayes

F. Galliano (AIM) Astromind 2019, CEA/Saclay 15 / 36 He Grew-up near Caen (Normandie), son of a cider producer& small estate owner. His father pushes him towards a religious career ⇒ studies theology at the University of Caen. Abandons this idea and moves to Paris ⇒ meets d’Alembert in 1769, who helps him to get a teaching position at the École Royale Militaire. Applies to the Académie Royale des Sciences ⇒ elected in 1773.

Pierre Simon (Hahn, 2004; English LAPLACE version in 2005) (1749–1827)

Upbringing & Early Career:

Laplace: the Probability of Causes of Events (1/2)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 16 / 36 He Grew-up near Caen (Normandie), son of a cider producer& small estate owner. His father pushes him towards a religious career ⇒ studies theology at the University of Caen. Abandons this idea and moves to Paris ⇒ meets d’Alembert in 1769, who helps him to get a teaching position at the École Royale Militaire. Applies to the Académie Royale des Sciences ⇒ elected in 1773.

(Hahn, 2004; English version in 2005)

Upbringing & Early Career:

Laplace: the Probability of Causes of Events (1/2)

Pierre Simon LAPLACE (1749–1827)

Upbringing & Early Career:

Laplace: the Probability of Causes of Events (1/2)

Pierre Simon (Hahn, 2004; English LAPLACE version in 2005) (1749–1827)

Laplace: the Probability of Causes of Events (1/2)

Pierre Simon (Hahn, 2004; English LAPLACE version in 2005) (1749–1827)

Upbringing & Early Career:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 16 / 36 His father pushes him towards a religious career ⇒ studies theology at the University of Caen. Abandons this idea and moves to Paris ⇒ meets d’Alembert in 1769, who helps him to get a teaching position at the École Royale Militaire. Applies to the Académie Royale des Sciences ⇒ elected in 1773.

Laplace: the Probability of Causes of Events (1/2)

Pierre Simon (Hahn, 2004; English LAPLACE version in 2005) (1749–1827)

Upbringing & Early Career: He Grew-up near Caen (Normandie), son of a cider producer& small estate owner.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 16 / 36 Abandons this idea and moves to Paris ⇒ meets d’Alembert in 1769, who helps him to get a teaching position at the École Royale Militaire. Applies to the Académie Royale des Sciences ⇒ elected in 1773.

Laplace: the Probability of Causes of Events (1/2)

Pierre Simon (Hahn, 2004; English LAPLACE version in 2005) (1749–1827)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 16 / 36 Applies to the Académie Royale des Sciences ⇒ elected in 1773.

Laplace: the Probability of Causes of Events (1/2)

Pierre Simon (Hahn, 2004; English LAPLACE version in 2005) (1749–1827)

Upbringing & Early Career: He Grew-up near Caen (Normandie), son of a cider producer& small estate owner. His father pushes him towards a religious career ⇒ studies theology at the University of Caen. Abandons this idea and moves to Paris ⇒ meets d’Alembert in 1769, who helps him to get a teaching position at the École Royale Militaire.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 16 / 36 Laplace: the Probability of Causes of Events (1/2)

Pierre Simon (Hahn, 2004; English LAPLACE version in 2005) (1749–1827)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 16 / 36 First application of Bayes’ rule (law of succession) ⇒ should say Bayesian-Laplacian method. He had not read Bayes memoir ⇒ discovered it in 1781, when Price went to Paris. Laplace acknowledged Bayes.

Newton’s geometric approach of 1799: (republic) interior minister. mechanics ⇒ analytic approach. 1808: (empire) Comte d’Empire. Origin & stability of the Solar system. 1817: (monarchy) Marquis. Postulated existence of black holes. 1814-1827: member of parliament.

Read Abraham DE MOIVRE’s memoir ⇒ understood probabilities can be used to quantify observational errors. Wrote: Mémoire sur la probabilité des causes par les événements (1774), where he rediscovered Bayes’ rule.

Studied demography ⇒ birth rate of males higher than females is a general rule of human kind.

(...) la théorie des probabilités n’est, au fond, que le bon sens réduit au calcul; elle fait apprécier avec exactitude ce que les esprits justes sentent par une sorte d’instinct, sans qu’ils puissent souvent s’en rendre compte. (Théorie analytique des probabilités, 1812)

Achievements in Celestial Mechanics: Political Career:

Laplace: the Probability of Causes of Events (2/2)

Contribution to the Theory of Probabilities:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 17 / 36 First application of Bayes’ rule (law of succession) ⇒ should say Bayesian-Laplacian method. He had not read Bayes memoir ⇒ discovered it in 1781, when Price went to Paris. Laplace acknowledged Bayes.

Wrote: Mémoire sur la probabilité des causes par les événements (1774), where he rediscovered Bayes’ rule.

Studied demography ⇒ birth rate of males higher than females is a general rule of human kind.

Achievements in Celestial Mechanics: Political Career:

Laplace: the Probability of Causes of Events (2/2)

Contribution to the Theory of Probabilities: Read Abraham DE MOIVRE’s memoir ⇒ understood probabilities can be used to quantify observational errors.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 17 / 36 Newton’s geometric approach of 1799: (republic) interior minister. mechanics ⇒ analytic approach. 1808: (empire) Comte d’Empire. Origin & stability of the Solar system. 1817: (monarchy) Marquis. Postulated existence of black holes. 1814-1827: member of parliament.

First application of Bayes’ rule (law of succession) ⇒ should say Bayesian-Laplacian method. He had not read Bayes memoir ⇒ discovered it in 1781, when Price went to Paris. Laplace acknowledged Bayes. Studied demography ⇒ birth rate of males higher than females is a general rule of human kind.

Achievements in Celestial Mechanics: Political Career:

Laplace: the Probability of Causes of Events (2/2)

He had not read Bayes memoir ⇒ discovered it in 1781, when Price went to Paris. Laplace acknowledged Bayes. Studied demography ⇒ birth rate of males higher than females is a general rule of human kind.

Achievements in Celestial Mechanics: Political Career:

Laplace: the Probability of Causes of Events (2/2)

Laplace acknowledged Bayes. Studied demography ⇒ birth rate of males higher than females is a general rule of human kind.

Achievements in Celestial Mechanics: Political Career:

Laplace: the Probability of Causes of Events (2/2)

Contribution to the Theory of Probabilities: Read Abraham DE MOIVRE’s memoir ⇒ understood probabilities can be used to quantify observational errors. Wrote: Mémoire sur la probabilité des causes par les événements (1774), where he rediscovered Bayes’ rule. First application of Bayes’ rule (law of succession) ⇒ should say Bayesian-Laplacian method. He had not read Bayes memoir ⇒ discovered it in 1781, when Price went to Paris.

Studied demography ⇒ birth rate of males higher than females is a general rule of human kind.

Achievements in Celestial Mechanics: Political Career:

Laplace: the Probability of Causes of Events (2/2)

Achievements in Celestial Mechanics: Political Career:

Laplace: the Probability of Causes of Events (2/2)

Achievements in Celestial Mechanics: Political Career:

Laplace: the Probability of Causes of Events (2/2)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 17 / 36 1799: (republic) interior minister. 1808: (empire) Comte d’Empire. 1817: (monarchy) Marquis. 1814-1827: member of parliament.

Political Career:

Newton’s geometric approach of mechanics ⇒ analytic approach. Origin & stability of the Solar system. Postulated existence of black holes.

Laplace: the Probability of Causes of Events (2/2)

Achievements in Celestial Mechanics:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 17 / 36 1799: (republic) interior minister. 1808: (empire) Comte d’Empire. 1817: (monarchy) Marquis. 1814-1827: member of parliament.

Political Career:

Origin & stability of the Solar system. Postulated existence of black holes.

Laplace: the Probability of Causes of Events (2/2)

Achievements in Celestial Mechanics: Newton’s geometric approach of mechanics ⇒ analytic approach.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 17 / 36 1799: (republic) interior minister. 1808: (empire) Comte d’Empire. 1817: (monarchy) Marquis. 1814-1827: member of parliament.

Political Career:

Postulated existence of black holes.

Laplace: the Probability of Causes of Events (2/2)

Achievements in Celestial Mechanics: Newton’s geometric approach of mechanics ⇒ analytic approach. Origin & stability of the Solar system.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 17 / 36 1799: (republic) interior minister. 1808: (empire) Comte d’Empire. 1817: (monarchy) Marquis. 1814-1827: member of parliament.

Political Career:

Laplace: the Probability of Causes of Events (2/2)

Achievements in Celestial Mechanics: Newton’s geometric approach of mechanics ⇒ analytic approach. Origin & stability of the Solar system. Postulated existence of black holes.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 17 / 36 1799: (republic) interior minister. 1808: (empire) Comte d’Empire. 1817: (monarchy) Marquis. 1814-1827: member of parliament.

Laplace: the Probability of Causes of Events (2/2)

Achievements in Celestial Mechanics: Political Career: Newton’s geometric approach of mechanics ⇒ analytic approach. Origin & stability of the Solar system. Postulated existence of black holes.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 17 / 36 1808: (empire) Comte d’Empire. 1817: (monarchy) Marquis. 1814-1827: member of parliament.

Laplace: the Probability of Causes of Events (2/2)

Achievements in Celestial Mechanics: Political Career: Newton’s geometric approach of 1799: (republic) interior minister. mechanics ⇒ analytic approach. Origin & stability of the Solar system. Postulated existence of black holes.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 17 / 36 1817: (monarchy) Marquis. 1814-1827: member of parliament.

Laplace: the Probability of Causes of Events (2/2)

Achievements in Celestial Mechanics: Political Career: Newton’s geometric approach of 1799: (republic) interior minister. mechanics ⇒ analytic approach. 1808: (empire) Comte d’Empire. Origin & stability of the Solar system. Postulated existence of black holes.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 17 / 36 1814-1827: member of parliament.

Laplace: the Probability of Causes of Events (2/2)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 17 / 36 Laplace: the Probability of Causes of Events (2/2)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 17 / 36 Mill initiated the frequentist approach 10 years after Laplace’s death. (...) a very slight improvement in the data, by better observations, or by taking into fuller consideration the special circumstances of the case, is of more use than the most elaborate application of the calculus of probabilities founded on the data in their previous state of inferiority. The neglect of this obvious reﬂection has given rise to misapplications of the calculus of probabilities which have made it the real opprobrium of mathematics. (A System of Logic, 1843) 1 The idea that probability should represent a degree of plausability seemed too vague to be the foundation for a mathematical theory& no clear way to assign priors. 2 The computation of Bayesian solutions was diﬃcult (no computers). 3 Laplace was despised in England for his Bonapartism.

John Stuart MILL Karl PEARSON (1806–1873) (1857–1936)

Rejection of Laplace’s Work:

After Laplace: Early Frequentist Development

F. Galliano (AIM) Astromind 2019, CEA/Saclay 18 / 36 Mill initiated the frequentist approach 10 years after Laplace’s death. (...) a very slight improvement in the data, by better observations, or by taking into fuller consideration the special circumstances of the case, is of more use than the most elaborate application of the calculus of probabilities founded on the data in their previous state of inferiority. The neglect of this obvious reﬂection has given rise to misapplications of the calculus of probabilities which have made it the real opprobrium of mathematics. (A System of Logic, 1843) 1 The idea that probability should represent a degree of plausability seemed too vague to be the foundation for a mathematical theory& no clear way to assign priors. 2 The computation of Bayesian solutions was diﬃcult (no computers). 3 Laplace was despised in England for his Bonapartism.

Karl PEARSON (1857–1936)

Rejection of Laplace’s Work:

After Laplace: Early Frequentist Development

John Stuart MILL (1806–1873)

Rejection of Laplace’s Work:

After Laplace: Early Frequentist Development

John Stuart MILL Karl PEARSON (1806–1873) (1857–1936)

After Laplace: Early Frequentist Development

John Stuart MILL Karl PEARSON (1806–1873) (1857–1936)

Rejection of Laplace’s Work:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 18 / 36 (...) a very slight improvement in the data, by better observations, or by taking into fuller consideration the special circumstances of the case, is of more use than the most elaborate application of the calculus of probabilities founded on the data in their previous state of inferiority. The neglect of this obvious reﬂection has given rise to misapplications of the calculus of probabilities which have made it the real opprobrium of mathematics. (A System of Logic, 1843) 1 The idea that probability should represent a degree of plausability seemed too vague to be the foundation for a mathematical theory& no clear way to assign priors. 2 The computation of Bayesian solutions was diﬃcult (no computers). 3 Laplace was despised in England for his Bonapartism.

After Laplace: Early Frequentist Development

John Stuart MILL Karl PEARSON (1806–1873) (1857–1936)

Rejection of Laplace’s Work: Mill initiated the frequentist approach 10 years after Laplace’s death.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 18 / 36 1 The idea that probability should represent a degree of plausability seemed too vague to be the foundation for a mathematical theory& no clear way to assign priors. 2 The computation of Bayesian solutions was diﬃcult (no computers). 3 Laplace was despised in England for his Bonapartism.

After Laplace: Early Frequentist Development

John Stuart MILL Karl PEARSON (1806–1873) (1857–1936)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 18 / 36 2 The computation of Bayesian solutions was diﬃcult (no computers). 3 Laplace was despised in England for his Bonapartism.

After Laplace: Early Frequentist Development

John Stuart MILL Karl PEARSON (1806–1873) (1857–1936)

Rejection of Laplace’s Work: Mill initiated the frequentist approach 10 years after Laplace’s death. (...) a very slight improvement in the data, by better observations, or by taking into fuller consideration the special circumstances of the case, is of more use than the most elaborate application of the calculus of probabilities founded on the data in their previous state of inferiority. The neglect of this obvious reﬂection has given rise to misapplications of the calculus of probabilities which have made it the real opprobrium of mathematics. (A System of Logic, 1843) 1 The idea that probability should represent a degree of plausability seemed too vague to be the foundation for a mathematical theory& no clear way to assign priors.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 18 / 36 3 Laplace was despised in England for his Bonapartism.

After Laplace: Early Frequentist Development

John Stuart MILL Karl PEARSON (1806–1873) (1857–1936)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 18 / 36 After Laplace: Early Frequentist Development

John Stuart MILL Karl PEARSON (1806–1873) (1857–1936)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 18 / 36 He was a social darwinist& eugenist. The Grammar of Science (1892) inﬂuenced Einstein. Developed: (1) χ2 test; (2) standard-deviation; (3) correlation; (4) p-value; (5) P.C.A. Initiator of the anti-Bayesian current.

Director of the department of eugenics at UCL. Developed: (1) maximum likelihood; (2) F-distribution/F-test; (3) null hypothesis & p < 0.05 (Statistical Methods for Research Workers, 1925). Paid as a consultant by the Tobacco Manufacturers’ Standing Committee ⇒ publicly against the 1950 study showing that tobacco causes lung cancer (“correlation does not imply causation”; British Medical Journal, 1957).

Karl PEARSON (1857–1936) Jerzy NEYMAN (1894–1981)

Sir Ronald FISHER (1890–1962)

Egon PEARSON (1895–1980)

The Golden Age of Frequentism (1920–1930)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 19 / 36 Director of the department of eugenics at UCL. Developed: (1) maximum likelihood; (2) F-distribution/F-test; (3) null hypothesis & p < 0.05 (Statistical Methods for Research Workers, 1925). Paid as a consultant by the Tobacco Manufacturers’ Standing Committee ⇒ publicly against the 1950 study showing that tobacco causes lung cancer (“correlation does not imply causation”; British Medical Journal, 1957).

Jerzy NEYMAN He was a social darwinist& eugenist. (1894–1981) The Grammar of Science (1892) inﬂuenced Einstein. Developed: (1) χ2 test; (2) standard-deviation; (3) correlation; (4) p-value; (5) P.C.A. Initiator of the anti-Bayesian current.

Sir Ronald FISHER (1890–1962)

Egon PEARSON (1895–1980)

The Golden Age of Frequentism (1920–1930)

Karl PEARSON (1857–1936)

Jerzy NEYMAN (1894–1981) The Grammar of Science (1892) inﬂuenced Einstein. Developed: (1) χ2 test; (2) standard-deviation; (3) correlation; (4) p-value; (5) P.C.A. Initiator of the anti-Bayesian current.

Sir Ronald FISHER (1890–1962)

Egon PEARSON (1895–1980)

The Golden Age of Frequentism (1920–1930)

Karl PEARSON (1857–1936)

He was a social darwinist& eugenist.

Jerzy NEYMAN (1894–1981)

Developed: (1) χ2 test; (2) standard-deviation; (3) correlation; (4) p-value; (5) P.C.A. Initiator of the anti-Bayesian current.

Sir Ronald FISHER (1890–1962)

Egon PEARSON (1895–1980)

The Golden Age of Frequentism (1920–1930)

Karl PEARSON (1857–1936)

He was a social darwinist& eugenist. The Grammar of Science (1892) inﬂuenced Einstein.

Jerzy NEYMAN (1894–1981)

Initiator of the anti-Bayesian current.

Sir Ronald FISHER (1890–1962)

Egon PEARSON (1895–1980)

The Golden Age of Frequentism (1920–1930)

Karl PEARSON (1857–1936)

He was a social darwinist& eugenist. The Grammar of Science (1892) inﬂuenced Einstein. Developed: (1) χ2 test; (2) standard-deviation; (3) correlation; (4) p-value; (5) P.C.A.

Jerzy NEYMAN (1894–1981)

Sir Ronald FISHER (1890–1962)

Egon PEARSON (1895–1980)

The Golden Age of Frequentism (1920–1930)

Karl PEARSON (1857–1936)

He was a social darwinist& eugenist. The Grammar of Science (1892) inﬂuenced Einstein. Developed: (1) χ2 test; (2) standard-deviation; (3) correlation; (4) p-value; (5) P.C.A. Initiator of the anti-Bayesian current.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 19 / 36 Jerzy NEYMAN (1894–1981)

Director of the department of eugenics at UCL. Egon PEARSON (1895–1980) Developed: (1) maximum likelihood; (2) F-distribution/F-test; (3) null hypothesis & p < 0.05 (Statistical Methods for Research Workers, 1925). Paid as a consultant by the Tobacco Manufacturers’ Standing Committee ⇒ publicly against the 1950 study showing that tobacco causes lung cancer (“correlation does not imply causation”; British Medical Journal, 1957).

The Golden Age of Frequentism (1920–1930)

Karl PEARSON (1857–1936)

Sir Ronald FISHER (1890–1962)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 19 / 36 Jerzy NEYMAN (1894–1981)

Egon PEARSON (1895–1980) Developed: (1) maximum likelihood; (2) F-distribution/F-test; (3) null hypothesis & p < 0.05 (Statistical Methods for Research Workers, 1925). Paid as a consultant by the Tobacco Manufacturers’ Standing Committee ⇒ publicly against the 1950 study showing that tobacco causes lung cancer (“correlation does not imply causation”; British Medical Journal, 1957).

The Golden Age of Frequentism (1920–1930)

Karl PEARSON (1857–1936)

Sir Ronald FISHER (1890–1962)

Director of the department of eugenics at UCL.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 19 / 36 Jerzy NEYMAN (1894–1981)

Egon PEARSON (1895–1980)

Paid as a consultant by the Tobacco Manufacturers’ Standing Committee ⇒ publicly against the 1950 study showing that tobacco causes lung cancer (“correlation does not imply causation”; British Medical Journal, 1957).

The Golden Age of Frequentism (1920–1930)

Karl PEARSON (1857–1936)

Sir Ronald FISHER (1890–1962)

Director of the department of eugenics at UCL. Developed: (1) maximum likelihood; (2) F-distribution/F-test; (3) null hypothesis & p < 0.05 (Statistical Methods for Research Workers, 1925).

F. Galliano (AIM) Astromind 2019, CEA/Saclay 19 / 36 Jerzy NEYMAN (1894–1981)

Egon PEARSON (1895–1980)

The Golden Age of Frequentism (1920–1930)

Karl PEARSON (1857–1936)

Sir Ronald FISHER (1890–1962)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 19 / 36 Egon PEARSON (1895–1980)

The Golden Age of Frequentism (1920–1930)

Karl PEARSON (1857–1936) Jerzy NEYMAN He was a social darwinist& eugenist. (1894–1981) The Grammar of Science (1892) inﬂuenced Einstein. Developed: (1) χ2 test; (2) standard-deviation; (3) correlation; (4) p-value; (5) P.C.A. Initiator of the anti-Bayesian current.

Sir Ronald FISHER (1890–1962)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 19 / 36 The Golden Age of Frequentism (1920–1930)

Sir Ronald FISHER (1890–1962)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 19 / 36 German submarines (U-boats) coded their In 1944, Thomas FLOWERS rused communication with cryptographic Enigma vacuum tubes instead of mechanical parts machines. ⇒ The Colossus computer. Turing built a mechanical computer (The Decodes Hitler’s intentions in case of Bomb) which tested the combinations. landing in Normandie ⇒ shorten the war He used Bayesian priors to reduce the by two years (Eisenhower). number of combinations, looking for The work at Bletchley Park remained frequent words, such as “ein”, “Heil classiﬁed until 1973, to hide that the Hitler” & meteorological terms. British could crack Russian codes He developed a unit quantifying the ⇒ delayed glory for the Bayesian weight of evidence (Bayes factor), named approach. the ban, after the city of Banbury, where punch cards were printed.

Alan TURING (1912–1954) Founder of theoretical computer science& artiﬁcial intelligence. Formalised the concepts of algorithm, computability & Turing machine.

The Enigma Code Breaking: The End of the War:

Bayes at Bletchley Park

F. Galliano (AIM) Astromind 2019, CEA/Saclay 20 / 36 German submarines (U-boats) coded their In 1944, Thomas FLOWERS rused communication with cryptographic Enigma vacuum tubes instead of mechanical parts machines. ⇒ The Colossus computer. Turing built a mechanical computer (The Decodes Hitler’s intentions in case of Bomb) which tested the combinations. landing in Normandie ⇒ shorten the war He used Bayesian priors to reduce the by two years (Eisenhower). number of combinations, looking for The work at Bletchley Park remained frequent words, such as “ein”, “Heil classiﬁed until 1973, to hide that the Hitler” & meteorological terms. British could crack Russian codes He developed a unit quantifying the ⇒ delayed glory for the Bayesian weight of evidence (Bayes factor), named approach. the ban, after the city of Banbury, where punch cards were printed.

Founder of theoretical computer science& artiﬁcial intelligence. Formalised the concepts of algorithm, computability & Turing machine.

The Enigma Code Breaking: The End of the War:

Bayes at Bletchley Park

Alan TURING (1912–1954)

Formalised the concepts of algorithm, computability & Turing machine.

The Enigma Code Breaking: The End of the War:

Bayes at Bletchley Park

Alan TURING (1912–1954) Founder of theoretical computer science& artiﬁcial intelligence.

The Enigma Code Breaking: The End of the War:

Bayes at Bletchley Park

Alan TURING (1912–1954) Founder of theoretical computer science& artiﬁcial intelligence. Formalised the concepts of algorithm, computability & Turing machine.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 20 / 36 In 1944, Thomas FLOWERS rused vacuum tubes instead of mechanical parts ⇒ The Colossus computer. Decodes Hitler’s intentions in case of landing in Normandie ⇒ shorten the war by two years (Eisenhower). The work at Bletchley Park remained classiﬁed until 1973, to hide that the British could crack Russian codes ⇒ delayed glory for the Bayesian approach.

The End of the War: German submarines (U-boats) coded their communication with cryptographic Enigma machines. Turing built a mechanical computer (The Bomb) which tested the combinations. He used Bayesian priors to reduce the number of combinations, looking for frequent words, such as “ein”, “Heil Hitler” & meteorological terms. He developed a unit quantifying the weight of evidence (Bayes factor), named the ban, after the city of Banbury, where punch cards were printed.

Bayes at Bletchley Park

Alan TURING (1912–1954) Founder of theoretical computer science& artiﬁcial intelligence. Formalised the concepts of algorithm, computability & Turing machine.

The Enigma Code Breaking:

The End of the War:

Turing built a mechanical computer (The Bomb) which tested the combinations. He used Bayesian priors to reduce the number of combinations, looking for frequent words, such as “ein”, “Heil Hitler” & meteorological terms. He developed a unit quantifying the weight of evidence (Bayes factor), named the ban, after the city of Banbury, where punch cards were printed.

Bayes at Bletchley Park

Alan TURING (1912–1954) Founder of theoretical computer science& artiﬁcial intelligence. Formalised the concepts of algorithm, computability & Turing machine.

The Enigma Code Breaking: German submarines (U-boats) coded their communication with cryptographic Enigma machines.

The End of the War:

He used Bayesian priors to reduce the number of combinations, looking for frequent words, such as “ein”, “Heil Hitler” & meteorological terms. He developed a unit quantifying the weight of evidence (Bayes factor), named the ban, after the city of Banbury, where punch cards were printed.

Bayes at Bletchley Park

Alan TURING (1912–1954) Founder of theoretical computer science& artiﬁcial intelligence. Formalised the concepts of algorithm, computability & Turing machine.

The Enigma Code Breaking: German submarines (U-boats) coded their communication with cryptographic Enigma machines. Turing built a mechanical computer (The Bomb) which tested the combinations.

The End of the War:

He developed a unit quantifying the weight of evidence (Bayes factor), named the ban, after the city of Banbury, where punch cards were printed.

Bayes at Bletchley Park

Alan TURING (1912–1954) Founder of theoretical computer science& artiﬁcial intelligence. Formalised the concepts of algorithm, computability & Turing machine.

The Enigma Code Breaking: German submarines (U-boats) coded their communication with cryptographic Enigma machines. Turing built a mechanical computer (The Bomb) which tested the combinations. He used Bayesian priors to reduce the number of combinations, looking for frequent words, such as “ein”, “Heil Hitler” & meteorological terms.

The End of the War:

Bayes at Bletchley Park

Alan TURING (1912–1954) Founder of theoretical computer science& artiﬁcial intelligence. Formalised the concepts of algorithm, computability & Turing machine.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 20 / 36 Decodes Hitler’s intentions in case of landing in Normandie ⇒ shorten the war by two years (Eisenhower). The work at Bletchley Park remained classiﬁed until 1973, to hide that the British could crack Russian codes ⇒ delayed glory for the Bayesian approach.

Bayes at Bletchley Park

Alan TURING (1912–1954) Founder of theoretical computer science& artiﬁcial intelligence. Formalised the concepts of algorithm, computability & Turing machine.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 20 / 36 The work at Bletchley Park remained classiﬁed until 1973, to hide that the British could crack Russian codes ⇒ delayed glory for the Bayesian approach.

Bayes at Bletchley Park

Alan TURING (1912–1954) Founder of theoretical computer science& artiﬁcial intelligence. Formalised the concepts of algorithm, computability & Turing machine.

The Enigma Code Breaking: The End of the War: German submarines (U-boats) coded their In 1944, Thomas FLOWERS rused communication with cryptographic Enigma vacuum tubes instead of mechanical parts machines. ⇒ The Colossus computer. Turing built a mechanical computer (The Decodes Hitler’s intentions in case of Bomb) which tested the combinations. landing in Normandie ⇒ shorten the war He used Bayesian priors to reduce the by two years (Eisenhower). number of combinations, looking for frequent words, such as “ein”, “Heil Hitler” & meteorological terms. He developed a unit quantifying the weight of evidence (Bayes factor), named the ban, after the city of Banbury, where punch cards were printed.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 20 / 36 Bayes at Bletchley Park

Alan TURING (1912–1954) Founder of theoretical computer science& artiﬁcial intelligence. Formalised the concepts of algorithm, computability & Turing machine.

The Enigma Code Breaking: The End of the War: German submarines (U-boats) coded their In 1944, Thomas FLOWERS rused communication with cryptographic Enigma vacuum tubes instead of mechanical parts machines. ⇒ The Colossus computer. Turing built a mechanical computer (The Decodes Hitler’s intentions in case of Bomb) which tested the combinations. landing in Normandie ⇒ shorten the war He used Bayesian priors to reduce the by two years (Eisenhower). number of combinations, looking for The work at Bletchley Park remained frequent words, such as “ein”, “Heil classiﬁed until 1973, to hide that the Hitler” & meteorological terms. British could crack Russian codes He developed a unit quantifying the ⇒ delayed glory for the Bayesian weight of evidence (Bayes factor), named approach. the ban, after the city of Banbury, where punch cards were printed.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 20 / 36 Geophysicist& mathematician (University of Cambridge). In 1926, Bayesian analysis of earthquakes (few data) ⇒ liquid Earth core. Initiated the Bayesian revival (Theory of Probability, 1939). Popularized the use of Bayes factors. Opposed to continental drift, though.

Arthur BAILEY applied Bayes’ rule, including the probability of events that had never happened ⇒ estimate insurance premiums. Dennis LINDLEY & Jimmy SAVAGE popularised Bayes’ rule (Savage: “Fisher is making Bayesian omelet without breaking Bayesian eggs”). Jerome CORNFIELD pioneered in applying Bayes’ rule to epidemiology ⇒ showed link between smoking & lung cancer (ridiculised Fisher). Howard RAIFFA & Robert SCHLAIFER taught Bayes’ rule for business& decision-making. Norman RASMUSSEN estimated the risks of a nuclear incident in the 70s ⇒ possible, but not necessarily desastrous (cf. Three Mile Island, 1979). Bayesian Search Algorithm used to ﬁnd lost nuclear bombs& russian submarines.

Sir Harold JEFFREYS (1891–1989)

Post World War II Era:

The Bayesian Resistance

F. Galliano (AIM) Astromind 2019, CEA/Saclay 21 / 36 Arthur BAILEY applied Bayes’ rule, including the probability of events that had never happened ⇒ estimate insurance premiums. Dennis LINDLEY & Jimmy SAVAGE popularised Bayes’ rule (Savage: “Fisher is making Bayesian omelet without breaking Bayesian eggs”). Jerome CORNFIELD pioneered in applying Bayes’ rule to epidemiology ⇒ showed link between smoking & lung cancer (ridiculised Fisher). Howard RAIFFA & Robert SCHLAIFER taught Bayes’ rule for business& decision-making. Norman RASMUSSEN estimated the risks of a nuclear incident in the 70s ⇒ possible, but not necessarily desastrous (cf. Three Mile Island, 1979). Bayesian Search Algorithm used to ﬁnd lost nuclear bombs& russian submarines.

Geophysicist& mathematician (University of Cambridge). In 1926, Bayesian analysis of earthquakes (few data) ⇒ liquid Earth core. Initiated the Bayesian revival (Theory of Probability, 1939). Popularized the use of Bayes factors. Opposed to continental drift, though.

Post World War II Era:

The Bayesian Resistance

Sir Harold JEFFREYS (1891–1989)

In 1926, Bayesian analysis of earthquakes (few data) ⇒ liquid Earth core. Initiated the Bayesian revival (Theory of Probability, 1939). Popularized the use of Bayes factors. Opposed to continental drift, though.

Post World War II Era:

The Bayesian Resistance

Sir Harold JEFFREYS (1891–1989)

Geophysicist& mathematician (University of Cambridge).

Initiated the Bayesian revival (Theory of Probability, 1939). Popularized the use of Bayes factors. Opposed to continental drift, though.

Post World War II Era:

The Bayesian Resistance

Sir Harold JEFFREYS (1891–1989)

Geophysicist& mathematician (University of Cambridge). In 1926, Bayesian analysis of earthquakes (few data) ⇒ liquid Earth core.

Popularized the use of Bayes factors. Opposed to continental drift, though.

Post World War II Era:

The Bayesian Resistance

Sir Harold JEFFREYS (1891–1989)

Geophysicist& mathematician (University of Cambridge). In 1926, Bayesian analysis of earthquakes (few data) ⇒ liquid Earth core. Initiated the Bayesian revival (Theory of Probability, 1939).

Opposed to continental drift, though.

Post World War II Era:

The Bayesian Resistance

Sir Harold JEFFREYS (1891–1989)

Post World War II Era:

The Bayesian Resistance

Sir Harold JEFFREYS (1891–1989)

The Bayesian Resistance

Sir Harold JEFFREYS (1891–1989)

Post World War II Era:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 21 / 36 Dennis LINDLEY & Jimmy SAVAGE popularised Bayes’ rule (Savage: “Fisher is making Bayesian omelet without breaking Bayesian eggs”). Jerome CORNFIELD pioneered in applying Bayes’ rule to epidemiology ⇒ showed link between smoking & lung cancer (ridiculised Fisher). Howard RAIFFA & Robert SCHLAIFER taught Bayes’ rule for business& decision-making. Norman RASMUSSEN estimated the risks of a nuclear incident in the 70s ⇒ possible, but not necessarily desastrous (cf. Three Mile Island, 1979). Bayesian Search Algorithm used to ﬁnd lost nuclear bombs& russian submarines.

The Bayesian Resistance

Sir Harold JEFFREYS (1891–1989)

Post World War II Era: Arthur BAILEY applied Bayes’ rule, including the probability of events that had never happened ⇒ estimate insurance premiums.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 21 / 36 Jerome CORNFIELD pioneered in applying Bayes’ rule to epidemiology ⇒ showed link between smoking & lung cancer (ridiculised Fisher). Howard RAIFFA & Robert SCHLAIFER taught Bayes’ rule for business& decision-making. Norman RASMUSSEN estimated the risks of a nuclear incident in the 70s ⇒ possible, but not necessarily desastrous (cf. Three Mile Island, 1979). Bayesian Search Algorithm used to ﬁnd lost nuclear bombs& russian submarines.

The Bayesian Resistance

Sir Harold JEFFREYS (1891–1989)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 21 / 36 Howard RAIFFA & Robert SCHLAIFER taught Bayes’ rule for business& decision-making. Norman RASMUSSEN estimated the risks of a nuclear incident in the 70s ⇒ possible, but not necessarily desastrous (cf. Three Mile Island, 1979). Bayesian Search Algorithm used to ﬁnd lost nuclear bombs& russian submarines.

The Bayesian Resistance

Sir Harold JEFFREYS (1891–1989)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 21 / 36 Norman RASMUSSEN estimated the risks of a nuclear incident in the 70s ⇒ possible, but not necessarily desastrous (cf. Three Mile Island, 1979). Bayesian Search Algorithm used to ﬁnd lost nuclear bombs& russian submarines.

The Bayesian Resistance

Sir Harold JEFFREYS (1891–1989)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 21 / 36 Bayesian Search Algorithm used to ﬁnd lost nuclear bombs& russian submarines.

The Bayesian Resistance

Sir Harold JEFFREYS (1891–1989)

Post World War II Era: Arthur BAILEY applied Bayes’ rule, including the probability of events that had never happened ⇒ estimate insurance premiums. Dennis LINDLEY & Jimmy SAVAGE popularised Bayes’ rule (Savage: “Fisher is making Bayesian omelet without breaking Bayesian eggs”). Jerome CORNFIELD pioneered in applying Bayes’ rule to epidemiology ⇒ showed link between smoking & lung cancer (ridiculised Fisher). Howard RAIFFA & Robert SCHLAIFER taught Bayes’ rule for business& decision-making. Norman RASMUSSEN estimated the risks of a nuclear incident in the 70s ⇒ possible, but not necessarily desastrous (cf. Three Mile Island, 1979).

F. Galliano (AIM) Astromind 2019, CEA/Saclay 21 / 36 The Bayesian Resistance

Sir Harold JEFFREYS (1891–1989)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 21 / 36 (1) very good agreement with theory of stellar collapse & neutron star formation; (2) upper limit on the mass of ν¯e .

1970: First Markov Chain Monte-Carlo (MCMC) method, the Metropolis-Hastings algorithm (Hastings, 1970). 1984: Most popular MCMC solver, the Gibbs sampling (Geman & Geman, 1984). ⇒ Bayesian techniques became more attractive.

1983: NASA estimated the probability of shuttle failure at 1 in 100 000, with frequentist methods. An independent Bayesian analysis estimated the odds of rocket booster failure at 1 in 35. In 1986, the 28th launch (Challenger) exploded. 1987: explosion of SN1987A, with two dozen neutrinos detected ⇒ Loredo & Lamb (1989) sucessfully applied Bayesian modelling, while frequentist techniques were failing to analyze this valuable data (Loredo, 1990, for a review):

2003: completion of the Human Genome Project, which used Bayesian techniques.

Most machine-learning techniques are probabilistic. Training a neural network ⇔ informing a prior.

Numerical Techniques to solve Bayesian Problems:

Modern Applications:

The Bayesian Foundation of Machine-Learning & A.I.:

The Great Numerical Leap Forward

F. Galliano (AIM) Astromind 2019, CEA/Saclay 22 / 36 (1) very good agreement with theory of stellar collapse & neutron star formation; (2) upper limit on the mass of ν¯e .

2003: completion of the Human Genome Project, which used Bayesian techniques.

Most machine-learning techniques are probabilistic. Training a neural network ⇔ informing a prior.

Modern Applications:

The Bayesian Foundation of Machine-Learning & A.I.:

The Great Numerical Leap Forward

Numerical Techniques to solve Bayesian Problems:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 22 / 36 (1) very good agreement with theory of stellar collapse & neutron star formation; (2) upper limit on the mass of ν¯e .

2003: completion of the Human Genome Project, which used Bayesian techniques.

Most machine-learning techniques are probabilistic. Training a neural network ⇔ informing a prior.

1984: Most popular MCMC solver, the Gibbs sampling (Geman & Geman, 1984). ⇒ Bayesian techniques became more attractive.

Modern Applications:

The Bayesian Foundation of Machine-Learning & A.I.:

The Great Numerical Leap Forward

Numerical Techniques to solve Bayesian Problems: 1970: First Markov Chain Monte-Carlo (MCMC) method, the Metropolis-Hastings algorithm (Hastings, 1970).

F. Galliano (AIM) Astromind 2019, CEA/Saclay 22 / 36 (1) very good agreement with theory of stellar collapse & neutron star formation; (2) upper limit on the mass of ν¯e .

2003: completion of the Human Genome Project, which used Bayesian techniques.

Most machine-learning techniques are probabilistic. Training a neural network ⇔ informing a prior.

⇒ Bayesian techniques became more attractive.

Modern Applications:

The Bayesian Foundation of Machine-Learning & A.I.:

The Great Numerical Leap Forward

Numerical Techniques to solve Bayesian Problems: 1970: First Markov Chain Monte-Carlo (MCMC) method, the Metropolis-Hastings algorithm (Hastings, 1970). 1984: Most popular MCMC solver, the Gibbs sampling (Geman & Geman, 1984).

F. Galliano (AIM) Astromind 2019, CEA/Saclay 22 / 36 (1) very good agreement with theory of stellar collapse & neutron star formation; (2) upper limit on the mass of ν¯e .

2003: completion of the Human Genome Project, which used Bayesian techniques.

Most machine-learning techniques are probabilistic. Training a neural network ⇔ informing a prior.

Modern Applications:

The Bayesian Foundation of Machine-Learning & A.I.:

The Great Numerical Leap Forward

F. Galliano (AIM) Astromind 2019, CEA/Saclay 22 / 36 (1) very good agreement with theory of stellar collapse & neutron star formation; (2) upper limit on the mass of ν¯e .

Most machine-learning techniques are probabilistic. Training a neural network ⇔ informing a prior.

2003: completion of the Human Genome Project, which used Bayesian techniques.

The Bayesian Foundation of Machine-Learning & A.I.:

The Great Numerical Leap Forward

Modern Applications:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 22 / 36 (1) very good agreement with theory of stellar collapse & neutron star formation; (2) upper limit on the mass of ν¯e .

Most machine-learning techniques are probabilistic. Training a neural network ⇔ informing a prior.

1987: explosion of SN1987A, with two dozen neutrinos detected ⇒ Loredo & Lamb (1989) sucessfully applied Bayesian modelling, while frequentist techniques were failing to analyze this valuable data (Loredo, 1990, for a review):

2003: completion of the Human Genome Project, which used Bayesian techniques.

The Bayesian Foundation of Machine-Learning & A.I.:

The Great Numerical Leap Forward

F. Galliano (AIM) Astromind 2019, CEA/Saclay 22 / 36 Most machine-learning techniques are probabilistic. Training a neural network ⇔ informing a prior.

(1) very good agreement with theory of stellar collapse & neutron star formation; (2) upper limit on the mass of ν¯e . 2003: completion of the Human Genome Project, which used Bayesian techniques.

The Bayesian Foundation of Machine-Learning & A.I.:

The Great Numerical Leap Forward

Modern Applications: 1983: NASA estimated the probability of shuttle failure at 1 in 100 000, with frequentist methods. An independent Bayesian analysis estimated the odds of rocket booster failure at 1 in 35. In 1986, the 28th launch (Challenger) exploded. 1987: explosion of SN1987A, with two dozen neutrinos detected ⇒ Loredo & Lamb (1989) sucessfully applied Bayesian modelling, while frequentist techniques were failing to analyze this valuable data (Loredo, 1990, for a review):

F. Galliano (AIM) Astromind 2019, CEA/Saclay 22 / 36 Most machine-learning techniques are probabilistic. Training a neural network ⇔ informing a prior.

2003: completion of the Human Genome Project, which used Bayesian techniques.

The Bayesian Foundation of Machine-Learning & A.I.:

The Great Numerical Leap Forward

F. Galliano (AIM) Astromind 2019, CEA/Saclay 22 / 36 Most machine-learning techniques are probabilistic. Training a neural network ⇔ informing a prior.

The Bayesian Foundation of Machine-Learning & A.I.:

The Great Numerical Leap Forward

F. Galliano (AIM) Astromind 2019, CEA/Saclay 22 / 36 Most machine-learning techniques are probabilistic. Training a neural network ⇔ informing a prior.

The Great Numerical Leap Forward

The Bayesian Foundation of Machine-Learning & A.I.:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 22 / 36 Training a neural network ⇔ informing a prior.

The Great Numerical Leap Forward

The Bayesian Foundation of Machine-Learning & A.I.: Most machine-learning techniques are probabilistic.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 22 / 36 The Great Numerical Leap Forward

The Bayesian Foundation of Machine-Learning & A.I.: Most machine-learning techniques are probabilistic. Training a neural network ⇔ informing a prior.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 22 / 36 (Lectures given between 2011 and 2012)

Importance of Bayes’ Rule For Neurosciences

F. Galliano (AIM) Astromind 2019, CEA/Saclay 23 / 36 Importance of Bayes’ Rule For Neurosciences

(Lectures given between 2011 and 2012)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 23 / 36 ⇒ the visual cortex interprets shades, using this prior.

(From Dehaene’s Lecture in College de France, 2011)

Most objects are illuminated from above (sunlight, spots, etc.)

Optical Illusions: Manifestation of the Brain’s Bayesian Prior

F. Galliano (AIM) Astromind 2019, CEA/Saclay 24 / 36 ⇒ the visual cortex interprets shades, using this prior.

Most objects are illuminated from above (sunlight, spots, etc.)

Optical Illusions: Manifestation of the Brain’s Bayesian Prior

(From Dehaene’s Lecture in College de France, 2011)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 24 / 36 ⇒ the visual cortex interprets shades, using this prior.

Most objects are illuminated from above (sunlight, spots, etc.)

Optical Illusions: Manifestation of the Brain’s Bayesian Prior

(From Dehaene’s Lecture in College de France, 2011)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 24 / 36 ⇒ the visual cortex interprets shades, using this prior.

Optical Illusions: Manifestation of the Brain’s Bayesian Prior

(From Dehaene’s Lecture in College de France, 2011)

Most objects are illuminated from above (sunlight, spots, etc.)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 24 / 36 Optical Illusions: Manifestation of the Brain’s Bayesian Prior

(From Dehaene’s Lecture in College de France, 2011)

Most objects are illuminated from above (sunlight, spots, etc.) ⇒ the visual cortex interprets shades, using this prior.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 24 / 36 (Tenenbaum et al., 2011)

How Does the Brain Learn New Words?

F. Galliano (AIM) Astromind 2019, CEA/Saclay 25 / 36 How Does the Brain Learn New Words?

(Tenenbaum et al., 2011)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 25 / 36 How Does the Brain Learn New Words?

(Tenenbaum et al., 2011)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 25 / 36 How Does the Brain Learn New Words?

(Tenenbaum et al., 2011)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 25 / 36 Outline of the Talk

1 BAYESIANS VS FREQUENTISTS Epistemological Principles & Comparison Demonstration on a Simple Example Limitations of the Frequentist Approach

2 BAYES’ RULE THROUGH HISTORY Early Development The Frequentist Winter The Bayesian Renaissance

3 IMPLICATIONS FOR THE SCIENTIFIC METHOD Karl Popper’s Logic of Scientiﬁc Discovery Bayesian Epistemology How Researchers Actually Work

4 SUMMARY & CONCLUSION

F. Galliano (AIM) Astromind 2019, CEA/Saclay 26 / 36 Epistemology: the philosophical study of the nature, origin, and limits of human knowledge. Deduction: inferring the truth of a speciﬁc case from general rules. Induction: inferring a general conclusion based on individual cases.

Aim at demarcating Human intuitions about itself from theology& the physical world are metaphysics. possibly ﬂawed (e.g. Euclidian geometry). Renounce to understand the causes (why), to Abandon rationalism for focus on the empiricism ⇒ a mathematical laws of proposition has a nature (how). cognitive meaning only if it can be veriﬁed by experience.

Preliminary Deﬁnitions:

Scientiﬁc Positivism: Conventionalism & Veriﬁcationism:

Auguste COMTE Henri (1798–1857) POINCARÉ (1854–1912)

The Epistemological Debate at the Beginning of the XXth Century

The unity of science consists alone in its method, not in its material. Karl PEARSON (The Grammar of Science, 1892)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 27 / 36 Aim at demarcating Human intuitions about itself from theology& the physical world are metaphysics. possibly ﬂawed (e.g. Euclidian geometry). Renounce to understand the causes (why), to Abandon rationalism for focus on the empiricism ⇒ a mathematical laws of proposition has a nature (how). cognitive meaning only if it can be veriﬁed by experience.

Epistemology: the philosophical study of the nature, origin, and limits of human knowledge. Deduction: inferring the truth of a speciﬁc case from general rules. Induction: inferring a general conclusion based on individual cases.

Scientiﬁc Positivism: Conventionalism & Veriﬁcationism:

Auguste COMTE Henri (1798–1857) POINCARÉ (1854–1912)

The Epistemological Debate at the Beginning of the XXth Century

The unity of science consists alone in its method, not in its material. Karl PEARSON (The Grammar of Science, 1892)

Preliminary Deﬁnitions:

Deduction: inferring the truth of a speciﬁc case from general rules. Induction: inferring a general conclusion based on individual cases.

Scientiﬁc Positivism: Conventionalism & Veriﬁcationism:

Auguste COMTE Henri (1798–1857) POINCARÉ (1854–1912)

The Epistemological Debate at the Beginning of the XXth Century

The unity of science consists alone in its method, not in its material. Karl PEARSON (The Grammar of Science, 1892)

Preliminary Deﬁnitions: Epistemology: the philosophical study of the nature, origin, and limits of human knowledge.

Induction: inferring a general conclusion based on individual cases.

Scientiﬁc Positivism: Conventionalism & Veriﬁcationism:

Auguste COMTE Henri (1798–1857) POINCARÉ (1854–1912)

The Epistemological Debate at the Beginning of the XXth Century

The unity of science consists alone in its method, not in its material. Karl PEARSON (The Grammar of Science, 1892)

Preliminary Deﬁnitions: Epistemology: the philosophical study of the nature, origin, and limits of human knowledge. Deduction: inferring the truth of a speciﬁc case from general rules.

Scientiﬁc Positivism: Conventionalism & Veriﬁcationism:

Auguste COMTE Henri (1798–1857) POINCARÉ (1854–1912)

The Epistemological Debate at the Beginning of the XXth Century

The unity of science consists alone in its method, not in its material. Karl PEARSON (The Grammar of Science, 1892)

Preliminary Deﬁnitions: Epistemology: the philosophical study of the nature, origin, and limits of human knowledge. Deduction: inferring the truth of a speciﬁc case from general rules. Induction: inferring a general conclusion based on individual cases.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 27 / 36 Human intuitions about the physical world are possibly ﬂawed (e.g. Euclidian geometry). Abandon rationalism for empiricism ⇒ a proposition has a cognitive meaning only if it can be veriﬁed by experience.

Conventionalism & Veriﬁcationism:

Aim at demarcating itself from theology& metaphysics. Renounce to understand the causes (why), to focus on the mathematical laws of Henri nature (how). POINCARÉ (1854–1912)

The Epistemological Debate at the Beginning of the XXth Century

The unity of science consists alone in its method, not in its material. Karl PEARSON (The Grammar of Science, 1892)

Scientiﬁc Positivism:

Auguste COMTE (1798–1857)

Conventionalism & Veriﬁcationism:

Renounce to understand the causes (why), to focus on the mathematical laws of Henri nature (how). POINCARÉ (1854–1912)

The Epistemological Debate at the Beginning of the XXth Century

The unity of science consists alone in its method, not in its material. Karl PEARSON (The Grammar of Science, 1892)

Scientiﬁc Positivism:

Aim at demarcating itself from theology& metaphysics.

Auguste COMTE (1798–1857)

Conventionalism & Veriﬁcationism:

Henri POINCARÉ (1854–1912)

The Epistemological Debate at the Beginning of the XXth Century

The unity of science consists alone in its method, not in its material. Karl PEARSON (The Grammar of Science, 1892)

Scientiﬁc Positivism:

Aim at demarcating itself from theology& metaphysics. Renounce to understand the causes (why), to Auguste focus on the COMTE mathematical laws of (1798–1857) nature (how).

The Epistemological Debate at the Beginning of the XXth Century

The unity of science consists alone in its method, not in its material. Karl PEARSON (The Grammar of Science, 1892)

Scientiﬁc Positivism: Conventionalism & Veriﬁcationism:

Aim at demarcating itself from theology& metaphysics. Renounce to understand the causes (why), to Auguste focus on the COMTE mathematical laws of Henri (1798–1857) nature (how). POINCARÉ (1854–1912)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 27 / 36 Abandon rationalism for empiricism ⇒ a proposition has a cognitive meaning only if it can be veriﬁed by experience.

The Epistemological Debate at the Beginning of the XXth Century

The unity of science consists alone in its method, not in its material. Karl PEARSON (The Grammar of Science, 1892)

Scientiﬁc Positivism: Conventionalism & Veriﬁcationism:

Aim at demarcating Human intuitions about itself from theology& the physical world are metaphysics. possibly ﬂawed (e.g. Euclidian geometry). Renounce to understand the causes (why), to Auguste focus on the COMTE mathematical laws of Henri (1798–1857) nature (how). POINCARÉ (1854–1912)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 27 / 36 The Epistemological Debate at the Beginning of the XXth Century

The unity of science consists alone in its method, not in its material. Karl PEARSON (The Grammar of Science, 1892)

Scientiﬁc Positivism: Conventionalism & Veriﬁcationism:

Aim at demarcating Human intuitions about itself from theology& the physical world are metaphysics. possibly ﬂawed (e.g. Euclidian geometry). Renounce to understand the causes (why), to Abandon rationalism for empiricism ⇒ a Auguste focus on the proposition has a COMTE mathematical laws of Henri cognitive meaning only (1798–1857) nature (how). POINCARÉ (1854–1912) if it can be veriﬁed by experience.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 27 / 36 Inductive logic “does not provide a Criticize conventionalism who “evade suitable criterion of demarcation with falsification by using ad hoc modifications metaphysical speculation”. of the theory.” “The actual procedure of science is to “One must not save from falsification a operate with conjectures: to jump to theory if it has failed.” conclusions – often after one single ⇒ Falsifiabilism: deductivism& modus tollens. observation”. Modus tollens: (A ⇒ B) ∧ B ⇒ A. ⇒ Deductivism: “Hypotheses can only be emprically tested and only after they have been advanced”.

Karl POPPER Very inﬂuential book, still a (1902–1994) reference today (published in 1935; English version in 1959).

The Myth of Induction: Falsiﬁability, the Criterion of Demarcation:

Popper’s Criticism of Induction in Empirical Sciences

F. Galliano (AIM) Astromind 2019, CEA/Saclay 28 / 36 Inductive logic “does not provide a Criticize conventionalism who “evade suitable criterion of demarcation with falsification by using ad hoc modifications metaphysical speculation”. of the theory.” “The actual procedure of science is to “One must not save from falsification a operate with conjectures: to jump to theory if it has failed.” conclusions – often after one single ⇒ Falsifiabilism: deductivism& modus tollens. observation”. Modus tollens: (A ⇒ B) ∧ B ⇒ A. ⇒ Deductivism: “Hypotheses can only be emprically tested and only after they have been advanced”.

Very inﬂuential book, still a reference today (published in 1935; English version in 1959).

The Myth of Induction: Falsiﬁability, the Criterion of Demarcation:

Popper’s Criticism of Induction in Empirical Sciences

Karl POPPER (1902–1994)

The Myth of Induction: Falsiﬁability, the Criterion of Demarcation:

Popper’s Criticism of Induction in Empirical Sciences

Karl POPPER Very inﬂuential book, still a (1902–1994) reference today (published in 1935; English version in 1959).

F. Galliano (AIM) Astromind 2019, CEA/Saclay 28 / 36 Criticize conventionalism who “evade falsification by using ad hoc modifications of the theory.” “One must not save from falsification a theory if it has failed.” ⇒ Falsifiabilism: deductivism& modus tollens. Modus tollens: (A ⇒ B) ∧ B ⇒ A.

Falsiﬁability, the Criterion of Demarcation: Inductive logic “does not provide a suitable criterion of demarcation with metaphysical speculation”. “The actual procedure of science is to operate with conjectures: to jump to conclusions – often after one single observation”. ⇒ Deductivism: “Hypotheses can only be emprically tested and only after they have been advanced”.

Popper’s Criticism of Induction in Empirical Sciences

Karl POPPER Very inﬂuential book, still a (1902–1994) reference today (published in 1935; English version in 1959).

The Myth of Induction:

Falsiﬁability, the Criterion of Demarcation:

“The actual procedure of science is to operate with conjectures: to jump to conclusions – often after one single observation”. ⇒ Deductivism: “Hypotheses can only be emprically tested and only after they have been advanced”.

Popper’s Criticism of Induction in Empirical Sciences

Karl POPPER Very inﬂuential book, still a (1902–1994) reference today (published in 1935; English version in 1959).

The Myth of Induction: Inductive logic “does not provide a suitable criterion of demarcation with metaphysical speculation”.

Falsiﬁability, the Criterion of Demarcation:

⇒ Deductivism: “Hypotheses can only be emprically tested and only after they have been advanced”.

Popper’s Criticism of Induction in Empirical Sciences

Karl POPPER Very inﬂuential book, still a (1902–1994) reference today (published in 1935; English version in 1959).

The Myth of Induction: Inductive logic “does not provide a suitable criterion of demarcation with metaphysical speculation”. “The actual procedure of science is to operate with conjectures: to jump to conclusions – often after one single observation”.

Falsiﬁability, the Criterion of Demarcation:

Popper’s Criticism of Induction in Empirical Sciences

Karl POPPER Very inﬂuential book, still a (1902–1994) reference today (published in 1935; English version in 1959).

Popper’s Criticism of Induction in Empirical Sciences

Karl POPPER Very inﬂuential book, still a (1902–1994) reference today (published in 1935; English version in 1959).

The Myth of Induction: Falsiﬁability, the Criterion of Demarcation: Inductive logic “does not provide a suitable criterion of demarcation with metaphysical speculation”. “The actual procedure of science is to operate with conjectures: to jump to conclusions – often after one single observation”. ⇒ Deductivism: “Hypotheses can only be emprically tested and only after they have been advanced”.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 28 / 36 “One must not save from falsiﬁcation a theory if it has failed.” ⇒ Falsiﬁabilism: deductivism& modus tollens. Modus tollens: (A ⇒ B) ∧ B ⇒ A.

Popper’s Criticism of Induction in Empirical Sciences

Karl POPPER Very inﬂuential book, still a (1902–1994) reference today (published in 1935; English version in 1959).

The Myth of Induction: Falsifiability, the Criterion of Demarcation: Inductive logic “does not provide a Criticize conventionalism who “evade suitable criterion of demarcation with falsification by using ad hoc modifications metaphysical speculation”. of the theory.” “The actual procedure of science is to operate with conjectures: to jump to conclusions – often after one single observation”. ⇒ Deductivism: “Hypotheses can only be emprically tested and only after they have been advanced”.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 28 / 36 ⇒ Falsiﬁabilism: deductivism& modus tollens. Modus tollens: (A ⇒ B) ∧ B ⇒ A.

Popper’s Criticism of Induction in Empirical Sciences

Karl POPPER Very inﬂuential book, still a (1902–1994) reference today (published in 1935; English version in 1959).

The Myth of Induction: Falsifiability, the Criterion of Demarcation: Inductive logic “does not provide a Criticize conventionalism who “evade suitable criterion of demarcation with falsification by using ad hoc modifications metaphysical speculation”. of the theory.” “The actual procedure of science is to “One must not save from falsification a operate with conjectures: to jump to theory if it has failed.” conclusions – often after one single observation”. ⇒ Deductivism: “Hypotheses can only be emprically tested and only after they have been advanced”.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 28 / 36 Popper’s Criticism of Induction in Empirical Sciences

Karl POPPER Very inﬂuential book, still a (1902–1994) reference today (published in 1935; English version in 1959).

The Myth of Induction: Falsifiability, the Criterion of Demarcation: Inductive logic “does not provide a Criticize conventionalism who “evade suitable criterion of demarcation with falsification by using ad hoc modifications metaphysical speculation”. of the theory.” “The actual procedure of science is to “One must not save from falsification a operate with conjectures: to jump to theory if it has failed.” conclusions – often after one single ⇒ Falsifiabilism: deductivism& modus tollens. observation”. Modus tollens: (A ⇒ B) ∧ B ⇒ A. ⇒ Deductivism: “Hypotheses can only be emprically tested and only after they have been advanced”.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 28 / 36 The objectivity of scientiﬁc statements lie in the fact that they can be inter-subjectively tested. Only repeatable experiments can be tested by anyone ⇒ no coincidence.

Simplicity ⇔ degree of falsiﬁability ⇔ higher prior improbability. ⇒ simpler theories have a higher empirical content.

Falsifiability ⇔ p-value to reject the null hypothesis. Boolean/Platonic logic & reject of probability as quantification of knowledge (no account for its subjectivity). Necessary repeatability: no possibility to account for sparse, unique constraints (must first think about a falsifiable experience). Parsimony: ad hoc principle. Opposed to induction: “The logic of probable inference leads to apriorism”. ⇒ It was conceived at the peak of the frequentist winter.

Objectivity & Reproducibility:

Principle of Parsimony:

Popper’s Epistemology has a Frequentist Frame of Mind:

Objectivity & Principle of Parsimony (Occam’s Razor)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 29 / 36 Simplicity ⇔ degree of falsiﬁability ⇔ higher prior improbability. ⇒ simpler theories have a higher empirical content.

The objectivity of scientiﬁc statements lie in the fact that they can be inter-subjectively tested. Only repeatable experiments can be tested by anyone ⇒ no coincidence.

Principle of Parsimony:

Popper’s Epistemology has a Frequentist Frame of Mind:

Objectivity & Principle of Parsimony (Occam’s Razor)

Objectivity & Reproducibility:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 29 / 36 Simplicity ⇔ degree of falsiﬁability ⇔ higher prior improbability. ⇒ simpler theories have a higher empirical content.

Only repeatable experiments can be tested by anyone ⇒ no coincidence.

Principle of Parsimony:

Popper’s Epistemology has a Frequentist Frame of Mind:

Objectivity & Principle of Parsimony (Occam’s Razor)

Objectivity & Reproducibility: The objectivity of scientiﬁc statements lie in the fact that they can be inter-subjectively tested.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 29 / 36 Simplicity ⇔ degree of falsiﬁability ⇔ higher prior improbability. ⇒ simpler theories have a higher empirical content.

Principle of Parsimony:

Popper’s Epistemology has a Frequentist Frame of Mind:

Objectivity & Principle of Parsimony (Occam’s Razor)

Objectivity & Reproducibility: The objectivity of scientiﬁc statements lie in the fact that they can be inter-subjectively tested. Only repeatable experiments can be tested by anyone ⇒ no coincidence.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 29 / 36 Falsifiability ⇔ p-value to reject the null hypothesis. Boolean/Platonic logic & reject of probability as quantification of knowledge (no account for its subjectivity). Necessary repeatability: no possibility to account for sparse, unique constraints (must first think about a falsifiable experience). Parsimony: ad hoc principle. Opposed to induction: “The logic of probable inference leads to apriorism”. ⇒ It was conceived at the peak of the frequentist winter.

Simplicity ⇔ degree of falsiﬁability ⇔ higher prior improbability. ⇒ simpler theories have a higher empirical content.

Popper’s Epistemology has a Frequentist Frame of Mind:

Objectivity & Principle of Parsimony (Occam’s Razor)

Principle of Parsimony:

⇒ simpler theories have a higher empirical content.

Popper’s Epistemology has a Frequentist Frame of Mind:

Objectivity & Principle of Parsimony (Occam’s Razor)

Principle of Parsimony: Simplicity ⇔ degree of falsiﬁability ⇔ higher prior improbability.

Popper’s Epistemology has a Frequentist Frame of Mind:

Objectivity & Principle of Parsimony (Occam’s Razor)

Principle of Parsimony: Simplicity ⇔ degree of falsiﬁability ⇔ higher prior improbability. ⇒ simpler theories have a higher empirical content.

Objectivity & Principle of Parsimony (Occam’s Razor)

Principle of Parsimony: Simplicity ⇔ degree of falsiﬁability ⇔ higher prior improbability. ⇒ simpler theories have a higher empirical content.

Popper’s Epistemology has a Frequentist Frame of Mind:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 29 / 36 Boolean/Platonic logic & reject of probability as quantification of knowledge (no account for its subjectivity). Necessary repeatability: no possibility to account for sparse, unique constraints (must first think about a falsifiable experience). Parsimony: ad hoc principle. Opposed to induction: “The logic of probable inference leads to apriorism”. ⇒ It was conceived at the peak of the frequentist winter.

Objectivity & Principle of Parsimony (Occam’s Razor)

Principle of Parsimony: Simplicity ⇔ degree of falsiﬁability ⇔ higher prior improbability. ⇒ simpler theories have a higher empirical content.

Popper’s Epistemology has a Frequentist Frame of Mind: Falsiﬁability ⇔ p-value to reject the null hypothesis.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 29 / 36 Necessary repeatability: no possibility to account for sparse, unique constraints (must ﬁrst think about a falsiﬁable experience). Parsimony: ad hoc principle. Opposed to induction: “The logic of probable inference leads to apriorism”. ⇒ It was conceived at the peak of the frequentist winter.

Objectivity & Principle of Parsimony (Occam’s Razor)

Principle of Parsimony: Simplicity ⇔ degree of falsiﬁability ⇔ higher prior improbability. ⇒ simpler theories have a higher empirical content.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 29 / 36 Parsimony: ad hoc principle. Opposed to induction: “The logic of probable inference leads to apriorism”. ⇒ It was conceived at the peak of the frequentist winter.

Objectivity & Principle of Parsimony (Occam’s Razor)

Principle of Parsimony: Simplicity ⇔ degree of falsiﬁability ⇔ higher prior improbability. ⇒ simpler theories have a higher empirical content.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 29 / 36 Opposed to induction: “The logic of probable inference leads to apriorism”. ⇒ It was conceived at the peak of the frequentist winter.

Objectivity & Principle of Parsimony (Occam’s Razor)

Principle of Parsimony: Simplicity ⇔ degree of falsiﬁability ⇔ higher prior improbability. ⇒ simpler theories have a higher empirical content.

Popper’s Epistemology has a Frequentist Frame of Mind: Falsifiability ⇔ p-value to reject the null hypothesis. Boolean/Platonic logic & reject of probability as quantification of knowledge (no account for its subjectivity). Necessary repeatability: no possibility to account for sparse, unique constraints (must first think about a falsifiable experience). Parsimony: ad hoc principle.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 29 / 36 ⇒ It was conceived at the peak of the frequentist winter.

Objectivity & Principle of Parsimony (Occam’s Razor)

Principle of Parsimony: Simplicity ⇔ degree of falsiﬁability ⇔ higher prior improbability. ⇒ simpler theories have a higher empirical content.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 29 / 36 Objectivity & Principle of Parsimony (Occam’s Razor)

Principle of Parsimony: Simplicity ⇔ degree of falsiﬁability ⇔ higher prior improbability. ⇒ simpler theories have a higher empirical content.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 29 / 36 Sir Harold JEFFREYS (1891–1989)

Lê Nguyên HOANG (Published in 2003; (1987–) dedicated to Jeﬀreys) Edwin Thompson JAYNES (Published in (1922–1998) 2018)

A Bayesian Alternative to Popper’s Epistemology

F. Galliano (AIM) Astromind 2019, CEA/Saclay 30 / 36 Lê Nguyên HOANG (Published in 2003; (1987–) dedicated to Jeﬀreys) Edwin Thompson JAYNES (Published in (1922–1998) 2018)

A Bayesian Alternative to Popper’s Epistemology

Sir Harold JEFFREYS (1891–1989)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 30 / 36 Lê Nguyên HOANG (1987–)

(Published in 2018)

A Bayesian Alternative to Popper’s Epistemology

Sir Harold JEFFREYS (1891–1989)

(Published in 2003; dedicated to Jeﬀreys) Edwin Thompson JAYNES (1922–1998)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 30 / 36 A Bayesian Alternative to Popper’s Epistemology

Sir Harold JEFFREYS (1891–1989)

Lê Nguyên HOANG (Published in 2003; (1987–) dedicated to Jeﬀreys) Edwin Thompson JAYNES (Published in (1922–1998) 2018)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 30 / 36 ⇔ Not Black ⇒ Not a Raven | {z } contraposition

Raven ⇒ Black | {z } proposition

Thus: ⇒

1 − p(Black|Raven) p(Not a Raven|Not Black) = 1 − p(Raven) 6= p(Black|Raven) 1 − p(Black) Most importantly: the weight of evidence (Bayes factor) is small for a red apple (Good, 1960).

Hempel’s Paradox (Hempel, 1940):

Carl Gustav HEMPEL (1905–1997)

Bayesian Solution to the Paradox:

Raven ⇒ Black ⇔ p(Black|Raven) = 1

Popper’s Criterion of Demarcation & Bayesian Epistemology: No need to require falsiﬁcation ⇒ the weight of evidence tells us how relevant an observation is.

Falsiﬁability & the Limits of Platonic Logic

F. Galliano (AIM) Astromind 2019, CEA/Saclay 31 / 36 ⇔ Not Black ⇒ Not a Raven | {z } contraposition

1 − p(Black|Raven) p(Not a Raven|Not Black) = 1 − p(Raven) 6= p(Black|Raven) 1 − p(Black) Most importantly: the weight of evidence (Bayes factor) is small for a red apple (Good, 1960).

Raven ⇒ Black | {z } proposition

Thus: ⇒

Bayesian Solution to the Paradox:

Raven ⇒ Black ⇔ p(Black|Raven) = 1

Popper’s Criterion of Demarcation & Bayesian Epistemology: No need to require falsiﬁcation ⇒ the weight of evidence tells us how relevant an observation is.

Falsiﬁability & the Limits of Platonic Logic

Hempel’s Paradox (Hempel, 1940):

Carl Gustav HEMPEL (1905–1997)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 31 / 36 1 − p(Black|Raven) p(Not a Raven|Not Black) = 1 − p(Raven) 6= p(Black|Raven) 1 − p(Black) Most importantly: the weight of evidence (Bayes factor) is small for a red apple (Good, 1960).

⇔ Not Black ⇒ Not a Raven | {z } contraposition

Thus: ⇒

Bayesian Solution to the Paradox:

Raven ⇒ Black ⇔ p(Black|Raven) = 1

Popper’s Criterion of Demarcation & Bayesian Epistemology: No need to require falsiﬁcation ⇒ the weight of evidence tells us how relevant an observation is.

Falsiﬁability & the Limits of Platonic Logic

Hempel’s Paradox (Hempel, 1940):

Raven ⇒ Black | {z } proposition

Carl Gustav HEMPEL (1905–1997)

Thus: ⇒

Bayesian Solution to the Paradox:

Raven ⇒ Black ⇔ p(Black|Raven) = 1

Popper’s Criterion of Demarcation & Bayesian Epistemology: No need to require falsiﬁcation ⇒ the weight of evidence tells us how relevant an observation is.

Falsiﬁability & the Limits of Platonic Logic

Hempel’s Paradox (Hempel, 1940):

Raven ⇒ Black ⇔ Not Black ⇒ Not a Raven | {z } | {z } proposition contraposition

Carl Gustav HEMPEL (1905–1997)

Bayesian Solution to the Paradox:

Raven ⇒ Black ⇔ p(Black|Raven) = 1

Popper’s Criterion of Demarcation & Bayesian Epistemology: No need to require falsiﬁcation ⇒ the weight of evidence tells us how relevant an observation is.

Falsiﬁability & the Limits of Platonic Logic

Hempel’s Paradox (Hempel, 1940):

Raven ⇒ Black ⇔ Not Black ⇒ Not a Raven | {z } | {z } proposition contraposition

Carl Gustav HEMPEL (1905–1997) Thus: ⇒

Popper’s Criterion of Demarcation & Bayesian Epistemology: No need to require falsiﬁcation ⇒ the weight of evidence tells us how relevant an observation is.

Falsiﬁability & the Limits of Platonic Logic

Hempel’s Paradox (Hempel, 1940):

Raven ⇒ Black ⇔ Not Black ⇒ Not a Raven | {z } | {z } proposition contraposition

Carl Gustav HEMPEL (1905–1997) Thus: ⇒

Bayesian Solution to the Paradox:

Raven ⇒ Black ⇔ p(Black|Raven) = 1

F. Galliano (AIM) Astromind 2019, CEA/Saclay 31 / 36 6= p(Black|Raven)

Most importantly: the weight of evidence (Bayes factor) is small for a red apple (Good, 1960).

Popper’s Criterion of Demarcation & Bayesian Epistemology: No need to require falsiﬁcation ⇒ the weight of evidence tells us how relevant an observation is.

Falsiﬁability & the Limits of Platonic Logic

Hempel’s Paradox (Hempel, 1940):

Raven ⇒ Black ⇔ Not Black ⇒ Not a Raven | {z } | {z } proposition contraposition

Carl Gustav HEMPEL (1905–1997) Thus: ⇒

Bayesian Solution to the Paradox:

Raven ⇒ Black ⇔ p(Black|Raven) = 1 1 − p(Black|Raven) p(Not a Raven|Not Black) = 1 − p(Raven) 1 − p(Black)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 31 / 36 Most importantly: the weight of evidence (Bayes factor) is small for a red apple (Good, 1960).

Popper’s Criterion of Demarcation & Bayesian Epistemology: No need to require falsiﬁcation ⇒ the weight of evidence tells us how relevant an observation is.

Falsiﬁability & the Limits of Platonic Logic

Hempel’s Paradox (Hempel, 1940):

Raven ⇒ Black ⇔ Not Black ⇒ Not a Raven | {z } | {z } proposition contraposition

Carl Gustav HEMPEL (1905–1997) Thus: ⇒

Bayesian Solution to the Paradox:

Raven ⇒ Black ⇔ p(Black|Raven) = 1 1 − p(Black|Raven) p(Not a Raven|Not Black) = 1 − p(Raven) 6= p(Black|Raven) 1 − p(Black)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 31 / 36 Popper’s Criterion of Demarcation & Bayesian Epistemology: No need to require falsiﬁcation ⇒ the weight of evidence tells us how relevant an observation is.

Falsiﬁability & the Limits of Platonic Logic

Hempel’s Paradox (Hempel, 1940):

Raven ⇒ Black ⇔ Not Black ⇒ Not a Raven | {z } | {z } proposition contraposition

Carl Gustav HEMPEL (1905–1997) Thus: ⇒

Bayesian Solution to the Paradox:

Raven ⇒ Black ⇔ p(Black|Raven) = 1 1 − p(Black|Raven) p(Not a Raven|Not Black) = 1 − p(Raven) 6= p(Black|Raven) 1 − p(Black) Most importantly: the weight of evidence (Bayes factor) is small for a red apple (Good, 1960).

F. Galliano (AIM) Astromind 2019, CEA/Saclay 31 / 36 Falsiﬁability & the Limits of Platonic Logic

Hempel’s Paradox (Hempel, 1940):

Raven ⇒ Black ⇔ Not Black ⇒ Not a Raven | {z } | {z } proposition contraposition

Carl Gustav HEMPEL (1905–1997) Thus: ⇒

Bayesian Solution to the Paradox:

Popper’s Criterion of Demarcation & Bayesian Epistemology: No need to require falsiﬁcation ⇒ the weight of evidence tells us how relevant an observation is.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 31 / 36 Z p(M1|data) = p(data|x1) × p(x1|M1) dx1

1 ∝ L(x ˆ1)δx1 × ∆x1 ⇒ Bayes factor between models M1 (1 parameter) & M2 (2 parameters): p(M2|data) δx2 ∝ L(x ˆ2) × p(M1|data) ∆x2 | {z } 1

p(par) ∝ 1 × p(data(1)|par) × ... × p(data(N)|par) | {z } |{z} | {z } new prior initial prior cumulated data

Reproducibility is Useful, But Not Necessary: Multiple experiments increasing evidence, but single observations are meaningful.

The Parsimony Principle is Hard-Coded in Bayes Factors:

The Prior Allows Accumulation of Knowledge:

Reproducibility, Parsimony & Accumulation of Knowledge

F. Galliano (AIM) Astromind 2019, CEA/Saclay 32 / 36 Z p(M1|data) = p(data|x1) × p(x1|M1) dx1

1 ∝ L(x ˆ1)δx1 × ∆x1 ⇒ Bayes factor between models M1 (1 parameter) & M2 (2 parameters): p(M2|data) δx2 ∝ L(x ˆ2) × p(M1|data) ∆x2 | {z } 1

p(par) ∝ 1 × p(data(1)|par) × ... × p(data(N)|par) | {z } |{z} | {z } new prior initial prior cumulated data

The Parsimony Principle is Hard-Coded in Bayes Factors:

The Prior Allows Accumulation of Knowledge:

Reproducibility, Parsimony & Accumulation of Knowledge

Reproducibility is Useful, But Not Necessary: Multiple experiments increasing evidence, but single observations are meaningful.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 32 / 36 p(par) ∝ 1 × p(data(1)|par) × ... × p(data(N)|par) | {z } |{z} | {z } new prior initial prior cumulated data

Z p(M1|data) = p(data|x1) × p(x1|M1) dx1

1 ∝ L(x ˆ1)δx1 × ∆x1 ⇒ Bayes factor between models M1 (1 parameter) & M2 (2 parameters): p(M2|data) δx2 ∝ L(x ˆ2) × p(M1|data) ∆x2 | {z } 1

The Prior Allows Accumulation of Knowledge:

Reproducibility, Parsimony & Accumulation of Knowledge

Reproducibility is Useful, But Not Necessary: Multiple experiments increasing evidence, but single observations are meaningful.

The Parsimony Principle is Hard-Coded in Bayes Factors:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 32 / 36 p(par) ∝ 1 × p(data(1)|par) × ... × p(data(N)|par) | {z } |{z} | {z } new prior initial prior cumulated data

Z p(M1|data) = p(data|x1) × p(x1|M1) dx1

1 ∝ L(x ˆ1)δx1 × ∆x1 ⇒ Bayes factor between models M1 (1 parameter) & M2 (2 parameters): p(M2|data) δx2 ∝ L(x ˆ2) × p(M1|data) ∆x2 | {z } 1

The Prior Allows Accumulation of Knowledge:

Reproducibility, Parsimony & Accumulation of Knowledge

Reproducibility is Useful, But Not Necessary: Multiple experiments increasing evidence, but single observations are meaningful.

The Parsimony Principle is Hard-Coded in Bayes Factors:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 32 / 36 p(par) ∝ 1 × p(data(1)|par) × ... × p(data(N)|par) | {z } |{z} | {z } new prior initial prior cumulated data

Z p(M1|data) = p(data|x1) × p(x1|M1) dx1

1 ∝ L(x ˆ1)δx1 × ∆x1 ⇒ Bayes factor between models M1 (1 parameter) & M2 (2 parameters): p(M2|data) δx2 ∝ L(x ˆ2) × p(M1|data) ∆x2 | {z } 1

The Prior Allows Accumulation of Knowledge:

Reproducibility, Parsimony & Accumulation of Knowledge

Reproducibility is Useful, But Not Necessary: Multiple experiments increasing evidence, but single observations are meaningful.

The Parsimony Principle is Hard-Coded in Bayes Factors:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 32 / 36 p(par) ∝ 1 × p(data(1)|par) × ... × p(data(N)|par) | {z } |{z} | {z } new prior initial prior cumulated data

Z p(M1|data) = p(data|x1) × p(x1|M1) dx1

1 ∝ L(x ˆ1)δx1 × ∆x1 ⇒ Bayes factor between models M1 (1 parameter) & M2 (2 parameters): p(M2|data) δx2 ∝ L(x ˆ2) × p(M1|data) ∆x2 | {z } 1

The Prior Allows Accumulation of Knowledge:

Reproducibility, Parsimony & Accumulation of Knowledge

Reproducibility is Useful, But Not Necessary: Multiple experiments increasing evidence, but single observations are meaningful.

The Parsimony Principle is Hard-Coded in Bayes Factors:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 32 / 36 p(par) ∝ 1 × p(data(1)|par) × ... × p(data(N)|par) | {z } |{z} | {z } new prior initial prior cumulated data

Z p(M1|data) = p(data|x1) × p(x1|M1) dx1

1 ∝ L(x ˆ1)δx1 × ∆x1 ⇒ Bayes factor between models M1 (1 parameter) & M2 (2 parameters): p(M2|data) δx2 ∝ L(x ˆ2) × p(M1|data) ∆x2 | {z } 1

The Prior Allows Accumulation of Knowledge:

Reproducibility, Parsimony & Accumulation of Knowledge

Reproducibility is Useful, But Not Necessary: Multiple experiments increasing evidence, but single observations are meaningful.

The Parsimony Principle is Hard-Coded in Bayes Factors:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 32 / 36 p(par) ∝ 1 × p(data(1)|par) × ... × p(data(N)|par) | {z } |{z} | {z } new prior initial prior cumulated data

Z p(M1|data) = p(data|x1) × p(x1|M1) dx1

1 ∝ L(x ˆ1)δx1 × ∆x1 ⇒ Bayes factor between models M1 (1 parameter) & M2 (2 parameters): p(M2|data) δx2 ∝ L(x ˆ2) × p(M1|data) ∆x2 | {z } 1

The Prior Allows Accumulation of Knowledge:

Reproducibility, Parsimony & Accumulation of Knowledge

Reproducibility is Useful, But Not Necessary: Multiple experiments increasing evidence, but single observations are meaningful.

The Parsimony Principle is Hard-Coded in Bayes Factors:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 32 / 36 p(par) ∝ 1 × p(data(1)|par) × ... × p(data(N)|par) | {z } |{z} | {z } new prior initial prior cumulated data

Z p(data|x1) × p(x1|M1) dx1

1 ∝ L(x ˆ1)δx1 × ∆x1 ⇒ Bayes factor between models M1 (1 parameter) & M2 (2 parameters): p(M2|data) δx2 ∝ L(x ˆ2) × p(M1|data) ∆x2 | {z } 1

The Prior Allows Accumulation of Knowledge:

Reproducibility, Parsimony & Accumulation of Knowledge

Reproducibility is Useful, But Not Necessary: Multiple experiments increasing evidence, but single observations are meaningful.

The Parsimony Principle is Hard-Coded in Bayes Factors:

p(M1|data) =

F. Galliano (AIM) Astromind 2019, CEA/Saclay 32 / 36 p(par) ∝ 1 × p(data(1)|par) × ... × p(data(N)|par) | {z } |{z} | {z } new prior initial prior cumulated data

1 ∝ L(x ˆ1)δx1 × ∆x1 ⇒ Bayes factor between models M1 (1 parameter) & M2 (2 parameters): p(M2|data) δx2 ∝ L(x ˆ2) × p(M1|data) ∆x2 | {z } 1

The Prior Allows Accumulation of Knowledge:

Reproducibility, Parsimony & Accumulation of Knowledge

Reproducibility is Useful, But Not Necessary: Multiple experiments increasing evidence, but single observations are meaningful.

The Parsimony Principle is Hard-Coded in Bayes Factors: Z p(M1|data) = p(data|x1) × p(x1|M1) dx1

F. Galliano (AIM) Astromind 2019, CEA/Saclay 32 / 36 p(par) ∝ 1 × p(data(1)|par) × ... × p(data(N)|par) | {z } |{z} | {z } new prior initial prior cumulated data

∆x1 ⇒ Bayes factor between models M1 (1 parameter) & M2 (2 parameters): p(M2|data) δx2 ∝ L(x ˆ2) × p(M1|data) ∆x2 | {z } 1

The Prior Allows Accumulation of Knowledge:

Reproducibility, Parsimony & Accumulation of Knowledge

Reproducibility is Useful, But Not Necessary: Multiple experiments increasing evidence, but single observations are meaningful.

The Parsimony Principle is Hard-Coded in Bayes Factors: Z p(M1|data) = p(data|x1) × p(x1|M1) dx1

∝ L(x ˆ1)δx1 ×

F. Galliano (AIM) Astromind 2019, CEA/Saclay 32 / 36 p(par) ∝ 1 × p(data(1)|par) × ... × p(data(N)|par) | {z } |{z} | {z } new prior initial prior cumulated data

⇒ Bayes factor between models M1 (1 parameter) & M2 (2 parameters): p(M2|data) δx2 ∝ L(x ˆ2) × p(M1|data) ∆x2 | {z } 1

The Prior Allows Accumulation of Knowledge:

Reproducibility, Parsimony & Accumulation of Knowledge

Reproducibility is Useful, But Not Necessary: Multiple experiments increasing evidence, but single observations are meaningful.

The Parsimony Principle is Hard-Coded in Bayes Factors: Z p(M1|data) = p(data|x1) × p(x1|M1) dx1

1 ∝ L(x ˆ1)δx1 × ∆x1

F. Galliano (AIM) Astromind 2019, CEA/Saclay 32 / 36 p(par) ∝ 1 × p(data(1)|par) × ... × p(data(N)|par) | {z } |{z} | {z } new prior initial prior cumulated data

p(M2|data) δx2 ∝ L(x ˆ2) × p(M1|data) ∆x2 | {z } 1

The Prior Allows Accumulation of Knowledge:

Reproducibility, Parsimony & Accumulation of Knowledge

Reproducibility is Useful, But Not Necessary: Multiple experiments increasing evidence, but single observations are meaningful.

The Parsimony Principle is Hard-Coded in Bayes Factors: Z p(M1|data) = p(data|x1) × p(x1|M1) dx1

1 ∝ L(x ˆ1)δx1 × ∆x1 ⇒ Bayes factor between models M1 (1 parameter) & M2 (2 parameters):

F. Galliano (AIM) Astromind 2019, CEA/Saclay 32 / 36 p(par) ∝ 1 × p(data(1)|par) × ... × p(data(N)|par) | {z } |{z} | {z } new prior initial prior cumulated data

δx2 L(x ˆ2) × ∆x2 | {z } 1

The Prior Allows Accumulation of Knowledge:

Reproducibility, Parsimony & Accumulation of Knowledge

Reproducibility is Useful, But Not Necessary: Multiple experiments increasing evidence, but single observations are meaningful.

The Parsimony Principle is Hard-Coded in Bayes Factors: Z p(M1|data) = p(data|x1) × p(x1|M1) dx1

1 ∝ L(x ˆ1)δx1 × ∆x1 ⇒ Bayes factor between models M1 (1 parameter) & M2 (2 parameters): p(M |data) 2 ∝ p(M1|data)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 32 / 36 p(par) ∝ 1 × p(data(1)|par) × ... × p(data(N)|par) | {z } |{z} | {z } new prior initial prior cumulated data

The Prior Allows Accumulation of Knowledge:

Reproducibility, Parsimony & Accumulation of Knowledge

Reproducibility is Useful, But Not Necessary: Multiple experiments increasing evidence, but single observations are meaningful.

The Parsimony Principle is Hard-Coded in Bayes Factors: Z p(M1|data) = p(data|x1) × p(x1|M1) dx1

1 ∝ L(x ˆ1)δx1 × ∆x1 ⇒ Bayes factor between models M1 (1 parameter) & M2 (2 parameters): p(M2|data) δx2 ∝ L(x ˆ2) × p(M1|data) ∆x2 | {z } 1

F. Galliano (AIM) Astromind 2019, CEA/Saclay 32 / 36 p(par) ∝ 1 × p(data(1)|par) × ... × p(data(N)|par) | {z } |{z} | {z } new prior initial prior cumulated data

Reproducibility, Parsimony & Accumulation of Knowledge

Reproducibility is Useful, But Not Necessary: Multiple experiments increasing evidence, but single observations are meaningful.

The Parsimony Principle is Hard-Coded in Bayes Factors: Z p(M1|data) = p(data|x1) × p(x1|M1) dx1

1 ∝ L(x ˆ1)δx1 × ∆x1 ⇒ Bayes factor between models M1 (1 parameter) & M2 (2 parameters): p(M2|data) δx2 ∝ L(x ˆ2) × p(M1|data) ∆x2 | {z } 1

The Prior Allows Accumulation of Knowledge:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 32 / 36 ∝ 1 × p(data(1)|par) × ... × p(data(N)|par) |{z} | {z } initial prior cumulated data

Reproducibility, Parsimony & Accumulation of Knowledge

Reproducibility is Useful, But Not Necessary: Multiple experiments increasing evidence, but single observations are meaningful.

The Parsimony Principle is Hard-Coded in Bayes Factors: Z p(M1|data) = p(data|x1) × p(x1|M1) dx1

1 ∝ L(x ˆ1)δx1 × ∆x1 ⇒ Bayes factor between models M1 (1 parameter) & M2 (2 parameters): p(M2|data) δx2 ∝ L(x ˆ2) × p(M1|data) ∆x2 | {z } 1

The Prior Allows Accumulation of Knowledge:

p(par) | {z } new prior

F. Galliano (AIM) Astromind 2019, CEA/Saclay 32 / 36 p(data(1)|par) × ... × p(data(N)|par) | {z } cumulated data

Reproducibility, Parsimony & Accumulation of Knowledge

Reproducibility is Useful, But Not Necessary: Multiple experiments increasing evidence, but single observations are meaningful.

The Parsimony Principle is Hard-Coded in Bayes Factors: Z p(M1|data) = p(data|x1) × p(x1|M1) dx1

1 ∝ L(x ˆ1)δx1 × ∆x1 ⇒ Bayes factor between models M1 (1 parameter) & M2 (2 parameters): p(M2|data) δx2 ∝ L(x ˆ2) × p(M1|data) ∆x2 | {z } 1

The Prior Allows Accumulation of Knowledge:

p(par) ∝ 1 × | {z } |{z} new prior initial prior

F. Galliano (AIM) Astromind 2019, CEA/Saclay 32 / 36 Reproducibility, Parsimony & Accumulation of Knowledge

Reproducibility is Useful, But Not Necessary: Multiple experiments increasing evidence, but single observations are meaningful.

The Parsimony Principle is Hard-Coded in Bayes Factors: Z p(M1|data) = p(data|x1) × p(x1|M1) dx1

1 ∝ L(x ˆ1)δx1 × ∆x1 ⇒ Bayes factor between models M1 (1 parameter) & M2 (2 parameters): p(M2|data) δx2 ∝ L(x ˆ2) × p(M1|data) ∆x2 | {z } 1

The Prior Allows Accumulation of Knowledge:

p(par) ∝ 1 × p(data(1)|par) × ... × p(data(N)|par) | {z } |{z} | {z } new prior initial prior cumulated data

F. Galliano (AIM) Astromind 2019, CEA/Saclay 32 / 36 In 2011, it detected neutrinos appearing 1 + 2 × 10−5 times faster than light, at 6σ signiﬁcance (frequentist analysis; OPERA coll., 2012).

⇒ check your cables.

Oscillation Project with Emulsion-tRacking Apparatus (OPERA):

Strict Popperian/Frequentist: Bayesian:

⇒ reject special relativity. p(vν > c) 1

OPERA: an Experiment Refuted by Theory

F. Galliano (AIM) Astromind 2019, CEA/Saclay 33 / 36 ⇒ check your cables.

In 2011, it detected neutrinos appearing 1 + 2 × 10−5 times faster than light, at 6σ signiﬁcance (frequentist analysis; OPERA coll., 2012).

Strict Popperian/Frequentist: Bayesian:

⇒ reject special relativity. p(vν > c) 1

OPERA: an Experiment Refuted by Theory

Oscillation Project with Emulsion-tRacking Apparatus (OPERA):

F. Galliano (AIM) Astromind 2019, CEA/Saclay 33 / 36 ⇒ check your cables.

Strict Popperian/Frequentist: Bayesian:

⇒ reject special relativity. p(vν > c) 1

OPERA: an Experiment Refuted by Theory

Oscillation Project with Emulsion-tRacking Apparatus (OPERA): In 2011, it detected neutrinos appearing 1 + 2 × 10−5 times faster than light, at 6σ signiﬁcance (frequentist analysis; OPERA coll., 2012).

F. Galliano (AIM) Astromind 2019, CEA/Saclay 33 / 36 ⇒ check your cables.

Bayesian:

p(vν > c) 1

OPERA: an Experiment Refuted by Theory

Strict Popperian/Frequentist: ⇒ reject special relativity.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 33 / 36 ⇒ check your cables.

OPERA: an Experiment Refuted by Theory

Strict Popperian/Frequentist: Bayesian:

⇒ reject special relativity. p(vν > c) 1

F. Galliano (AIM) Astromind 2019, CEA/Saclay 33 / 36 OPERA: an Experiment Refuted by Theory

Strict Popperian/Frequentist: Bayesian:

⇒ reject special relativity. p(vν > c) 1 ⇒ check your cables.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 33 / 36 ⇒ no falsiﬁability.

1 Those two experiments are distinct. 2 An absence of detection would not have disproven general relativity

Those two experiments, combined, bring a large weight of evidence in favor of general relativity.

First Detection of Gravitational Waves: First Image of a Black Hole (M 87):

(LIGO coll., 2015) (Event Horizon coll., 2019)

Strict Popperian/Frequentist Point of View:

Bayesian Point of View:

Is Verifying General Relativity Useful?

F. Galliano (AIM) Astromind 2019, CEA/Saclay 34 / 36 ⇒ no falsiﬁability.

1 Those two experiments are distinct. 2 An absence of detection would not have disproven general relativity

Those two experiments, combined, bring a large weight of evidence in favor of general relativity.

First Image of a Black Hole (M 87):

(Event Horizon coll., 2019)

Strict Popperian/Frequentist Point of View:

Bayesian Point of View:

Is Verifying General Relativity Useful?

First Detection of Gravitational Waves:

(LIGO coll., 2015)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 34 / 36 ⇒ no falsiﬁability.

1 Those two experiments are distinct. 2 An absence of detection would not have disproven general relativity

Those two experiments, combined, bring a large weight of evidence in favor of general relativity.

Strict Popperian/Frequentist Point of View:

Bayesian Point of View:

Is Verifying General Relativity Useful?

First Detection of Gravitational Waves: First Image of a Black Hole (M 87):

(LIGO coll., 2015) (Event Horizon coll., 2019)

F. Galliano (AIM) Astromind 2019, CEA/Saclay 34 / 36 ⇒ no falsiﬁability.

Those two experiments, combined, bring a large weight of evidence in favor of general relativity.

1 Those two experiments are distinct. 2 An absence of detection would not have disproven general relativity

Bayesian Point of View:

Is Verifying General Relativity Useful?

First Detection of Gravitational Waves: First Image of a Black Hole (M 87):

(LIGO coll., 2015) (Event Horizon coll., 2019)

Strict Popperian/Frequentist Point of View:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 34 / 36 ⇒ no falsiﬁability.

Those two experiments, combined, bring a large weight of evidence in favor of general relativity.

2 An absence of detection would not have disproven general relativity

Bayesian Point of View:

Is Verifying General Relativity Useful?

First Detection of Gravitational Waves: First Image of a Black Hole (M 87):

(LIGO coll., 2015) (Event Horizon coll., 2019)

Strict Popperian/Frequentist Point of View:

1 Those two experiments are distinct.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 34 / 36 Those two experiments, combined, bring a large weight of evidence in favor of general relativity.

⇒ no falsiﬁability.

Bayesian Point of View:

Is Verifying General Relativity Useful?

First Detection of Gravitational Waves: First Image of a Black Hole (M 87):

(LIGO coll., 2015) (Event Horizon coll., 2019)

Strict Popperian/Frequentist Point of View:

1 Those two experiments are distinct. 2 An absence of detection would not have disproven general relativity

F. Galliano (AIM) Astromind 2019, CEA/Saclay 34 / 36 Those two experiments, combined, bring a large weight of evidence in favor of general relativity.

Bayesian Point of View:

Is Verifying General Relativity Useful?

First Detection of Gravitational Waves: First Image of a Black Hole (M 87):

(LIGO coll., 2015) (Event Horizon coll., 2019)

Strict Popperian/Frequentist Point of View:

1 Those two experiments are distinct. 2 An absence of detection would not have disproven general relativity ⇒ no falsiﬁability.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 34 / 36 Those two experiments, combined, bring a large weight of evidence in favor of general relativity.

Is Verifying General Relativity Useful?

First Detection of Gravitational Waves: First Image of a Black Hole (M 87):

(LIGO coll., 2015) (Event Horizon coll., 2019)

Strict Popperian/Frequentist Point of View:

1 Those two experiments are distinct. 2 An absence of detection would not have disproven general relativity ⇒ no falsiﬁability.

Bayesian Point of View:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 34 / 36 Is Verifying General Relativity Useful?

First Detection of Gravitational Waves: First Image of a Black Hole (M 87):

(LIGO coll., 2015) (Event Horizon coll., 2019)

Strict Popperian/Frequentist Point of View:

1 Those two experiments are distinct. 2 An absence of detection would not have disproven general relativity ⇒ no falsiﬁability.

Bayesian Point of View: Those two experiments, combined, bring a large weight of evidence in favor of general relativity.

F. Galliano (AIM) Astromind 2019, CEA/Saclay 34 / 36 Outline of the Talk

1 BAYESIANS VS FREQUENTISTS Epistemological Principles & Comparison Demonstration on a Simple Example Limitations of the Frequentist Approach

2 BAYES’ RULE THROUGH HISTORY Early Development The Frequentist Winter The Bayesian Renaissance

3 IMPLICATIONS FOR THE SCIENTIFIC METHOD Karl Popper’s Logic of Scientiﬁc Discovery Bayesian Epistemology How Researchers Actually Work

4 SUMMARY & CONCLUSION

F. Galliano (AIM) Astromind 2019, CEA/Saclay 35 / 36 Bayesian: probabilities quantify our knowledge; Frequentist: probabilities represent the frequency of occurence of a repeated event.

p-values can lead to inconsistent results; the method partitions knowledge; it is less ﬂexible than Bayesianism.

No need for falsiﬁability ⇒ weight of evidence quantiﬁes data relevance; No need for reproducibility ⇒ can account for sparser, unique constraints; Parsimony ⇒ Bayes factors; Allows accumulation of knowledge ⇒ informative prior; Allows probabilistic induction.

1 There are two competing epistemological conceptions of probability:

2 It appears there are fundamental issues with the frequentist approach:

3 After a century of frequentist supremacy, Bayesianism emerged victorious.

Popper’s influential scientific method relies on falsifiability & reproducibility. It has a frequentist frame of mind. Bayesian alternative addresses the problems of Popper’s epistemology:

This is already the way we think (at least qualitatively), because it is the way our brain works.

The Opposition Between Bayesianism & Frequentism:

Epistemological Insights from Bayesianism:

Summary & Conclusion

F. Galliano (AIM) Astromind 2019, CEA/Saclay 36 / 36 No need for falsiﬁability ⇒ weight of evidence quantiﬁes data relevance; No need for reproducibility ⇒ can account for sparser, unique constraints; Parsimony ⇒ Bayes factors; Allows accumulation of knowledge ⇒ informative prior; Allows probabilistic induction.

Bayesian: probabilities quantify our knowledge; Frequentist: probabilities represent the frequency of occurence of a repeated event.

p-values can lead to inconsistent results; the method partitions knowledge; it is less ﬂexible than Bayesianism.

Popper’s influential scientific method relies on falsifiability & reproducibility. It has a frequentist frame of mind. Bayesian alternative addresses the problems of Popper’s epistemology:

This is already the way we think (at least qualitatively), because it is the way our brain works.

1 There are two competing epistemological conceptions of probability:

2 It appears there are fundamental issues with the frequentist approach:

3 After a century of frequentist supremacy, Bayesianism emerged victorious.

Epistemological Insights from Bayesianism:

Summary & Conclusion

The Opposition Between Bayesianism & Frequentism:

p-values can lead to inconsistent results; the method partitions knowledge; it is less ﬂexible than Bayesianism.

Popper’s influential scientific method relies on falsifiability & reproducibility. It has a frequentist frame of mind. Bayesian alternative addresses the problems of Popper’s epistemology:

This is already the way we think (at least qualitatively), because it is the way our brain works.

Bayesian: probabilities quantify our knowledge; Frequentist: probabilities represent the frequency of occurence of a repeated event. 2 It appears there are fundamental issues with the frequentist approach:

3 After a century of frequentist supremacy, Bayesianism emerged victorious.

Epistemological Insights from Bayesianism:

Summary & Conclusion

The Opposition Between Bayesianism & Frequentism:

1 There are two competing epistemological conceptions of probability:

p-values can lead to inconsistent results; the method partitions knowledge; it is less ﬂexible than Bayesianism.

Popper’s influential scientific method relies on falsifiability & reproducibility. It has a frequentist frame of mind. Bayesian alternative addresses the problems of Popper’s epistemology:

This is already the way we think (at least qualitatively), because it is the way our brain works.

Frequentist: probabilities represent the frequency of occurence of a repeated event. 2 It appears there are fundamental issues with the frequentist approach:

3 After a century of frequentist supremacy, Bayesianism emerged victorious.

Epistemological Insights from Bayesianism:

Summary & Conclusion

The Opposition Between Bayesianism & Frequentism:

1 There are two competing epistemological conceptions of probability: Bayesian: probabilities quantify our knowledge;

p-values can lead to inconsistent results; the method partitions knowledge; it is less ﬂexible than Bayesianism.

Popper’s influential scientific method relies on falsifiability & reproducibility. It has a frequentist frame of mind. Bayesian alternative addresses the problems of Popper’s epistemology:

This is already the way we think (at least qualitatively), because it is the way our brain works.

2 It appears there are fundamental issues with the frequentist approach:

3 After a century of frequentist supremacy, Bayesianism emerged victorious.

Epistemological Insights from Bayesianism:

Summary & Conclusion

The Opposition Between Bayesianism & Frequentism:

Popper’s influential scientific method relies on falsifiability & reproducibility. It has a frequentist frame of mind. Bayesian alternative addresses the problems of Popper’s epistemology:

This is already the way we think (at least qualitatively), because it is the way our brain works.

p-values can lead to inconsistent results; the method partitions knowledge; it is less ﬂexible than Bayesianism. 3 After a century of frequentist supremacy, Bayesianism emerged victorious.

Epistemological Insights from Bayesianism:

Summary & Conclusion

The Opposition Between Bayesianism & Frequentism:

Popper’s influential scientific method relies on falsifiability & reproducibility. It has a frequentist frame of mind. Bayesian alternative addresses the problems of Popper’s epistemology:

This is already the way we think (at least qualitatively), because it is the way our brain works.

the method partitions knowledge; it is less ﬂexible than Bayesianism. 3 After a century of frequentist supremacy, Bayesianism emerged victorious.

Epistemological Insights from Bayesianism:

Summary & Conclusion

The Opposition Between Bayesianism & Frequentism:

Popper’s influential scientific method relies on falsifiability & reproducibility. It has a frequentist frame of mind. Bayesian alternative addresses the problems of Popper’s epistemology:

This is already the way we think (at least qualitatively), because it is the way our brain works.

it is less ﬂexible than Bayesianism. 3 After a century of frequentist supremacy, Bayesianism emerged victorious.

Epistemological Insights from Bayesianism:

Summary & Conclusion

The Opposition Between Bayesianism & Frequentism:

1 There are two competing epistemological conceptions of probability: Bayesian: probabilities quantify our knowledge; Frequentist: probabilities represent the frequency of occurence of a repeated event. 2 It appears there are fundamental issues with the frequentist approach: p-values can lead to inconsistent results; the method partitions knowledge;

Popper’s influential scientific method relies on falsifiability & reproducibility. It has a frequentist frame of mind. Bayesian alternative addresses the problems of Popper’s epistemology:

This is already the way we think (at least qualitatively), because it is the way our brain works.

3 After a century of frequentist supremacy, Bayesianism emerged victorious.

Epistemological Insights from Bayesianism:

Summary & Conclusion

The Opposition Between Bayesianism & Frequentism:

Popper’s influential scientific method relies on falsifiability & reproducibility. It has a frequentist frame of mind. Bayesian alternative addresses the problems of Popper’s epistemology:

This is already the way we think (at least qualitatively), because it is the way our brain works.

Epistemological Insights from Bayesianism:

Summary & Conclusion

The Opposition Between Bayesianism & Frequentism:

Popper’s influential scientific method relies on falsifiability & reproducibility. It has a frequentist frame of mind. Bayesian alternative addresses the problems of Popper’s epistemology:

This is already the way we think (at least qualitatively), because it is the way our brain works.

Summary & Conclusion

The Opposition Between Bayesianism & Frequentism:

Epistemological Insights from Bayesianism:

Bayesian alternative addresses the problems of Popper’s epistemology:

This is already the way we think (at least qualitatively), because it is the way our brain works.

Summary & Conclusion

The Opposition Between Bayesianism & Frequentism:

Epistemological Insights from Bayesianism: Popper’s influential scientific method relies on falsifiability & reproducibility. It has a frequentist frame of mind.

Summary & Conclusion

The Opposition Between Bayesianism & Frequentism:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 36 / 36 No need for reproducibility ⇒ can account for sparser, unique constraints; Parsimony ⇒ Bayes factors; Allows accumulation of knowledge ⇒ informative prior; Allows probabilistic induction. This is already the way we think (at least qualitatively), because it is the way our brain works.

Summary & Conclusion

The Opposition Between Bayesianism & Frequentism:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 36 / 36 Parsimony ⇒ Bayes factors; Allows accumulation of knowledge ⇒ informative prior; Allows probabilistic induction. This is already the way we think (at least qualitatively), because it is the way our brain works.

Summary & Conclusion

The Opposition Between Bayesianism & Frequentism:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 36 / 36 Allows accumulation of knowledge ⇒ informative prior; Allows probabilistic induction. This is already the way we think (at least qualitatively), because it is the way our brain works.

Summary & Conclusion

The Opposition Between Bayesianism & Frequentism:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 36 / 36 Allows probabilistic induction. This is already the way we think (at least qualitatively), because it is the way our brain works.

Summary & Conclusion

The Opposition Between Bayesianism & Frequentism:

Epistemological Insights from Bayesianism: Popper’s influential scientific method relies on falsifiability & reproducibility. It has a frequentist frame of mind. Bayesian alternative addresses the problems of Popper’s epistemology: No need for falsifiability ⇒ weight of evidence quantifies data relevance; No need for reproducibility ⇒ can account for sparser, unique constraints; Parsimony ⇒ Bayes factors; Allows accumulation of knowledge ⇒ informative prior;

F. Galliano (AIM) Astromind 2019, CEA/Saclay 36 / 36 This is already the way we think (at least qualitatively), because it is the way our brain works.

Summary & Conclusion

The Opposition Between Bayesianism & Frequentism:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 36 / 36 Summary & Conclusion

The Opposition Between Bayesianism & Frequentism:

F. Galliano (AIM) Astromind 2019, CEA/Saclay 36 / 36