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) = = .