Belief Propagation in [.1Cm] Bayesian Networks

Belief Propagation in Bayesian Networks

Céline Comte

Reading Group “Network Theory” November 5, 2018 Introduction

(First-order) logic Represent causal relations between variables by a directed acyclic graph

Probabilities Weight these causal relations by probabilities that implicitly account for non-represented variables

“Belief networks are directed acyclic graphs in which the nodes represent propositions (or variables), the arcs signify direct dependencies between the linked propositions, and the strengths of these dependencies are quantified by conditional probabilities” (Pearl, 1986)

• A memory-efficient way of storing a PMF • Based on simple probability rules (more details in a few slides) • Inspired by human causal reasoning (Pearl, 1986, 1988) • Used for decision taking if a utility function is provided • Applied in many fields: medecine diagnoses, turbo-codes, (programming) language detection, … • Related to other models: Markov random fields, Markov chains, hidden Markov models, …

• J. Pearl (1982). “Reverend Bayes on Inference Engines: A Distributed Hierarchical Approach”. In: AAAI’82 → Belief propagation in causal trees

• J. Pearl (1986). “Fusion, propagation, and structuring in belief networks”. In: Artificial Intelligence → Belief propagation in causal trees and polytrees

• J. Pearl (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann → A complete reference (thanks Achille for providing me with this book)

• T. D. Nielsen and F. V. Jensen (2007). Bayesian Networks and Decision Graphs. Springer-Verlag → A lot of examples in Chapters 2 and 3

• D. Koller, N. Friedman, and F. Bach (2009). Probabilistic Graphical Models: Principles and Techniques. MIT Press

• M. Jordan (Last modified in 2015). An Introduction to Probabilistic Graphical Models. → Definition and belief probagation (thanks Nathan for pointing this reference)

Reminders on probability theory

Bayesian networks

Belief propagation in trees

Belief propagation in polytrees

Reminders on probability theory

Bayesian networks

Belief propagation in trees

Belief propagation in polytrees

• A and B are conditonally independent given C (written A ⊥⊥ B | C) if one of these three equivalent conditions is satisfied: − P(A,B | C) = P(A | C)P(B | C) − P(A | B,C) = P(A | C) − P(B | A,C) = P(B | C)

Independence and conditional independence

Remark: We work exclusively with discrete random variables

8/40 © 2018 Nokia Public • A and B are conditonally independent given C (written A ⊥⊥ B | C) if one of these three equivalent conditions is satisfied: − P(A,B | C) = P(A | C)P(B | C) − P(A | B,C) = P(A | C) − P(B | A,C) = P(B | C)

Independence and conditional independence

Remark: We work exclusively with discrete random variables

• A and B are marginally independent (written A ⊥⊥ B) if one of these three equivalent conditions is satisfied: − P(A,B) = P(A)P(B) − P(A | B) = P(A) − P(B | A) = P(B)

Remark: We work exclusively with discrete random variables

• A and B are marginally independent (written A ⊥⊥ B) if one of these three equivalent conditions is satisfied: − P(A,B) = P(A)P(B) − P(A | B) = P(A) − P(B | A) = P(B)

Useful rules

• The chain rule of probabilities If A1,...,An are random variables, we have

P(A1,...,An) = P(A1) × P(A2 | A1) × P(A3 | A1,A2) × ··· × P(An | A1,...,An−1)

• The chain rule of probabilities If A1,...,An are random variables, we have

P(A1,...,An) = P(A1) × P(A2 | A1) × P(A3 | A1,A2) × ··· × P(An | A1,...,An−1)

• Law of total probability If A and B are two random variables, ∑ P(B) = P(B | A)P(A) A

• Bayes’ rule If A and B are two random variables,

P(A | B)P(B) P(B | A) = P(A)

We can see P(A) as a normalizing constant: we can first compute P(B | A) ∝ P(A | B)P(B) for each value of B and then normalize to obtain P(B | A) without computing P(A)

• Evidence (piece of evidence) The set of random variables that have been observed

Glossary

• Belief in a random variable (conviction in french) Marginal distribution of this random variable (given the value of some observed variables)

Glossary

• Belief in a random variable (conviction in french) Marginal distribution of this random variable (given the value of some observed variables)

• Observe a random variable

• Belief in a random variable (conviction in french) Marginal distribution of this random variable (given the value of some observed variables)

• Observe a random variable

• Evidence (piece of evidence) The set of random variables that have been observed

Reminders on probability theory

Bayesian networks

Belief propagation in trees

Belief propagation in polytrees

G B

The Student example

Course Student difficulty intelligence

Grade Baccalauréat

Reference letter

Borrowed from (Koller, Friedman, and Bach, 2009)

G B Course Student difficulty intelligence L

Grade Baccalauréat

Reference letter

Borrowed from (Koller, Friedman, and Bach, 2009)

These two definitions are equivalent

The Student example D I

• Local Markov property G B Each node is conditionally independent of its non-descendants given its parents: L

D ⊥⊥ {I,B}, I ⊥⊥ D, G ⊥⊥ B | {D,I}, B ⊥⊥ {D,G,L} | I, L ⊥⊥ {D,I,B} | G

The Student example D I

• Local Markov property G B Each node is conditionally independent of its non-descendants given its parents: L

D ⊥⊥ {I,B}, I ⊥⊥ D, G ⊥⊥ B | {D,I}, B ⊥⊥ {D,G,L} | I, L ⊥⊥ {D,I,B} | G

• Chain rule of Bayesian networks By the chain rule of probabilities:

• Local Markov property G B Each node is conditionally independent of its non-descendants given its parents: L

D ⊥⊥ {I,B}, I ⊥⊥ D, G ⊥⊥ B | {D,I}, B ⊥⊥ {D,G,L} | I, L ⊥⊥ {D,I,B} | G

• Chain rule of Bayesian networks By the chain rule of probabilities:

These two definitions are equivalent

G B

Course Student L P(D) P(I) difficulty intelligence

P(G | D,I) Grade Baccalauréat

P(B | I)

P(L | G) Reference letter

G B

Described by L

• A directed acyclic graph

− Nodes ∼ (discrete) random variables X1,...,Xn − Arrows ∼ conditional (in)dependencies • Local conditional probability tables (CPT)

− P(Xi | parents(Xi)) for each node Xi

G B Two equivalent definitions L • Local Markov property Each node is conditionally independent of its non-descendants given its parents • Chain rule of Bayesian networks

∏n P(X1,...,Xn) = P(Xi | parents(Xi)) i=1 Proof of the equivalence: Corollary 4 p.20 of (Pearl, 1988)

17/40 © 2018 Nokia Public • Interpretation: chain of causality X “causes” Y that “causes” Z y // • Information can flow between X and Z through Y (that is, observing X changes our belief in Z and vice versa),

Flow of information unless Y is observed • Example: Markov chains

Base case: serial connection

X ⊥⊥ Z | YP(X,Y,Z) = P(X)P(Y | X)P(Z | Y) X

Flow of information unless Y is observed • Example: Markov chains

Base case: serial connection

X ⊥⊥ Z | YP(X,Y,Z) = P(X)P(Y | X)P(Z | Y) X • Interpretation: chain of causality X “causes” Y that “causes” Z Y

• Example: Markov chains

Base case: serial connection

X ⊥⊥ Z | YP(X,Y,Z) = P(X)P(Y | X)P(Z | Y) X • Interpretation: chain of causality X “causes” Y that “causes” Z Y • Information can flow between X and Z through Y (that is, observing X changes our belief in Z and vice versa), Z unless Y is observed

• Example: Markov chains

Base case: serial connection

• Example: Markov chains

Base case: serial connection

X ⊥⊥ Z | YP(X,Y,Z) = P(X)P(Y | X)P(Z | Y) X • Interpretation: chain of causality X “causes” Y that “causes” Z y // • Information can flow between X and Z through Y (that is, observing X changes our belief in Z and vice versa), Z Flow of information unless Y is observed

Base case: serial connection

• Interpretation: a single root cause Y with two observable consequences X and Z • Information can flow between X and Z through Y, unless Y is observed

Base case: diverging connection

X Z

X ⊥⊥ Z | YP(X,Y,Z) = P(X | Y)P(Y)P(Z | Y)

• Information can flow between X and Z through Y, unless Y is observed

Base case: diverging connection

X Z

X ⊥⊥ Z | YP(X,Y,Z) = P(X | Y)P(Y)P(Z | Y)

• Interpretation: a single root cause Y with two observable consequences X and Z

Base case: diverging connection

X Z

X ⊥⊥ Z | YP(X,Y,Z) = P(X | Y)P(Y)P(Z | Y)

• Interpretation: a single root cause Y with two observable consequences X and Z • Information can flow between X and Z through Y, unless Y is observed

Base case: diverging connection

X Z

X ⊥⊥ Z | YP(X,Y,Z) = P(X | Y)P(Y)P(Z | Y)

• Interpretation: a single root cause Y with two observable consequences X and Z • Information can flow between X and Z through Y, unless Y is observed

Base case: diverging connection

y //

X Z

X ⊥⊥ Z | YP(X,Y,Z) = P(X | Y)P(Y)P(Z | Y)

• Interpretation: a single root cause Y with two observable consequences X and Z • Information can flow between X and Z through Y, unless Y is observed

• Interpretation: two possible explanations X and Z for an observed consequence Y • “Explaining away” effect: information cannot flow between X and Z, unless Y is observed

Base case: converging connection

X Z

X ⊥⊥ ZP(X,Y,Z) = P(X)P(Y | X,Z)P(Z)

• “Explaining away” effect: information cannot flow between X and Z, unless Y is observed

Base case: converging connection

X Z

X ⊥⊥ ZP(X,Y,Z) = P(X)P(Y | X,Z)P(Z)

• Interpretation: two possible explanations X and Z for an observed consequence Y

Base case: converging connection

X Z

X ⊥⊥ ZP(X,Y,Z) = P(X)P(Y | X,Z)P(Z)

• Interpretation: two possible explanations X and Z for an observed consequence Y • “Explaining away” effect: information cannot flow between X and Z, unless Y is observed

Base case: converging connection

X Z // Y

X ⊥⊥ ZP(X,Y,Z) = P(X)P(Y | X,Z)P(Z)

• Interpretation: two possible explanations X and Z for an observed consequence Y • “Explaining away” effect: information cannot flow between X and Z, unless Y is observed

Base case: converging connection

X Z

X ⊥⊥ ZP(X,Y,Z) = P(X)P(Y | X,Z)P(Z)

• Interpretation: two possible explanations X and Z for an observed consequence Y • “Explaining away” effect: information cannot flow between X and Z, unless Y is observed

Which are correct? 1 G ⊥⊥ B? 2 B ⊥⊥ L? 3 D ⊥⊥ L? 4 D ⊥⊥ B? 5 D ⊥⊥ B | G? 6 D ⊥⊥ B | {I,G}?

Implied independencies D I

Similar to the “Strong Markov property” G B of Markov chains L

1 G ⊥⊥ B? 2 B ⊥⊥ L? 3 D ⊥⊥ L? 4 D ⊥⊥ B? 5 D ⊥⊥ B | G? 6 D ⊥⊥ B | {I,G}?

Implied independencies D I

Similar to the “Strong Markov property” G B of Markov chains L Which are correct?

No 2 B ⊥⊥ L? 3 D ⊥⊥ L? 4 D ⊥⊥ B? 5 D ⊥⊥ B | G? 6 D ⊥⊥ B | {I,G}?

Implied independencies D I

Similar to the “Strong Markov property” G B of Markov chains L Which are correct? 1 G ⊥⊥ B?

2 B ⊥⊥ L? 3 D ⊥⊥ L? 4 D ⊥⊥ B? 5 D ⊥⊥ B | G? 6 D ⊥⊥ B | {I,G}?

Implied independencies D I

Similar to the “Strong Markov property” G B of Markov chains L Which are correct? 1 G ⊥⊥ B? No

No 3 D ⊥⊥ L? 4 D ⊥⊥ B? 5 D ⊥⊥ B | G? 6 D ⊥⊥ B | {I,G}?

Implied independencies D I

Similar to the “Strong Markov property” G B of Markov chains L Which are correct? 1 G ⊥⊥ B? No 2 B ⊥⊥ L?

3 D ⊥⊥ L? 4 D ⊥⊥ B? 5 D ⊥⊥ B | G? 6 D ⊥⊥ B | {I,G}?

Implied independencies D I

Similar to the “Strong Markov property” G B of Markov chains L Which are correct? 1 G ⊥⊥ B? No 2 B ⊥⊥ L? No

No 4 D ⊥⊥ B? 5 D ⊥⊥ B | G? 6 D ⊥⊥ B | {I,G}?

Implied independencies D I

Similar to the “Strong Markov property” G B of Markov chains L Which are correct? 1 G ⊥⊥ B? No 2 B ⊥⊥ L? No 3 D ⊥⊥ L?

4 D ⊥⊥ B? 5 D ⊥⊥ B | G? 6 D ⊥⊥ B | {I,G}?

Implied independencies D I

Similar to the “Strong Markov property” G B of Markov chains L Which are correct? 1 G ⊥⊥ B? No 2 B ⊥⊥ L? No 3 D ⊥⊥ L? No

Yes (by the local Markov property applied to D) 5 D ⊥⊥ B | G? 6 D ⊥⊥ B | {I,G}?

Implied independencies D I

Similar to the “Strong Markov property” G B of Markov chains L Which are correct? 1 G ⊥⊥ B? No 2 B ⊥⊥ L? No 3 D ⊥⊥ L? No 4 D ⊥⊥ B?

5 D ⊥⊥ B | G? 6 D ⊥⊥ B | {I,G}?

Implied independencies D I

Similar to the “Strong Markov property” G B of Markov chains L Which are correct? 1 G ⊥⊥ B? No 2 B ⊥⊥ L? No 3 D ⊥⊥ L? No 4 D ⊥⊥ B? Yes (by the local Markov property applied to D)

No (“explaining away” effect) 6 D ⊥⊥ B | {I,G}?

Implied independencies D I

Similar to the “Strong Markov property” G B of Markov chains L Which are correct? 1 G ⊥⊥ B? No 2 B ⊥⊥ L? No 3 D ⊥⊥ L? No 4 D ⊥⊥ B? Yes (by the local Markov property applied to D) 5 D ⊥⊥ B | G?

6 D ⊥⊥ B | {I,G}?

Implied independencies D I

Similar to the “Strong Markov property” G B of Markov chains L Which are correct? 1 G ⊥⊥ B? No 2 B ⊥⊥ L? No 3 D ⊥⊥ L? No 4 D ⊥⊥ B? Yes (by the local Markov property applied to D) 5 D ⊥⊥ B | G? No (“explaining away” effect)

Implied independencies D I

Implied independencies D I

G B Proof of 6 D ⊥⊥ B | {I,G} L P(D,B | I,G)

Implied independencies D I

G B Proof of 6 D ⊥⊥ B | {I,G} L P(D,B | I,G)

= P(D | G,I)P(B | I,G)

Implied independencies D I

G B Proof of 6 D ⊥⊥ B | {I,G} ( ) L P(G | D,I,B)P(D,B | I) P(D,B | I,G) = Bayes’ P(G | I) rule

= P(D | G,I)P(B | I,G)

Implied independencies D I

G B Proof of 6 D ⊥⊥ B | {I,G} ( ) L P(G | D,I,B)P(D,B | I) P(D,B | I,G) = Bayes’ P(G | I) rule ( ) P(G | D,I)P(D,B | I) = local Markov property P(G | I) applied to G

= P(D | G,I)P(B | I,G)

Implied independencies D I

= P(D | G,I)P(B | I,G)

Implied independencies D I

G B Proof of 6 D ⊥⊥ B | {I,G} ( ) L P(G | D,I,B)P(D,B | I) P(D,B | I,G) = Bayes’ P(G | I) rule ( ) P(G | D,I)P(D,B | I) = local Markov property P(G | I) applied to G ( ) P(G | D,I)P(D | I)P(B | D,I) = definition of condi- P(G | I) tional probabilities P(G | D,I)P(D | I) = P(B | D,I) P(G | I)

= P(D | G,I)P(B | I,G)

Implied independencies D I

Memory complexity • If we store the probability distribution: O(rn) entries • If we store the node parents and the conditional ↑ ↑ probability tables: O(n(r + rd )) = O(nrd ) entries

What about the time complexity?

Memory and time complexity

Parameters

Memory complexity • If we store the probability distribution: O(rn) entries • If we store the node parents and the conditional ↑ ↑ probability tables: O(n(r + rd )) = O(nrd ) entries

What about the time complexity?

Memory and time complexity

Parameters n number of random variables (typically, n ∼ hundreds or thousands)

Memory complexity • If we store the probability distribution: O(rn) entries • If we store the node parents and the conditional ↑ ↑ probability tables: O(n(r + rd )) = O(nrd ) entries

What about the time complexity?

Memory and time complexity

Parameters n number of random variables (typically, n ∼ hundreds or thousands) r number of values each variable can take

23/40 © 2018 Nokia Public Memory complexity • If we store the probability distribution: O(rn) entries • If we store the node parents and the conditional ↑ ↑ probability tables: O(n(r + rd )) = O(nrd ) entries

What about the time complexity?

Memory and time complexity

Parameters n number of random variables (typically, n ∼ hundreds or thousands) r number of values each variable can take d↑ maximum number of parents of a node

What about the time complexity?

Memory and time complexity

Parameters n number of random variables (typically, n ∼ hundreds or thousands) r number of values each variable can take d↑ maximum number of parents of a node

Memory complexity

What about the time complexity?

Memory and time complexity

Parameters n number of random variables (typically, n ∼ hundreds or thousands) r number of values each variable can take d↑ maximum number of parents of a node

Memory complexity • If we store the probability distribution: O(rn) entries

Memory and time complexity

Parameters n number of random variables (typically, n ∼ hundreds or thousands) r number of values each variable can take d↑ maximum number of parents of a node

Memory complexity • If we store the probability distribution: O(rn) entries • If we store the node parents and the conditional ↑ ↑ probability tables: O(n(r + rd )) = O(nrd ) entries

Parameters n number of random variables (typically, n ∼ hundreds or thousands) r number of values each variable can take d↑ maximum number of parents of a node

Memory complexity • If we store the probability distribution: O(rn) entries • If we store the node parents and the conditional ↑ ↑ probability tables: O(n(r + rd )) = O(nrd ) entries

What about the time complexity?

Belief propagation, a.k.a. sum-product message passing: Propagate the information through the network, starting from the evidence node(s) • Each variable is a “separate processor” (a neuron?) that knows its own CPT and the messages received from its direct neighbors (Pearl, 1982) • Dynamic programming

Inference

“A guess that you make or an opinion that you form based on the information that you have” (Cambridge dictionary)

24/40 © 2018 Nokia Public Belief propagation, a.k.a. sum-product message passing: Propagate the information through the network, starting from the evidence node(s) • Each variable is a “separate processor” (a neuron?) that knows its own CPT and the messages received from its direct neighbors (Pearl, 1982) • Dynamic programming

Inference

“A guess that you make or an opinion that you form based on the information that you have” (Cambridge dictionary) → Bayesian networks: compute or update the belief in each variable given some evidence

Inference

Belief propagation, a.k.a. sum-product message passing: Propagate the information through the network, starting from the evidence node(s)

Inference

Reminders on probability theory

Bayesian networks

Belief propagation in trees

Belief propagation in polytrees

P(B | A) Each node separates the tree: its non-descendants and ( | ) ( | ) P C B P D B the subtrees rooted at each of its children are conditionally P(E | D) P(F |independentD) given this node

Remark: We will explain the propagation algorithm on this toy example borrowed from (Pearl, 1988)

Tree Bayesian network

A Each node (except the root) has at most one parent B

C D

E F

Remark: We will explain the propagation algorithm on this toy example borrowed from (Pearl, 1988)

Tree Bayesian network

P(A) A Each node (except the root) has at most one parent B P(B | A)

C P(C | B) D P(D | B)

P(E | D) E F P(F | D)

P(B | A) Each node separates the tree: its non-descendants and ( | ) ( | ) P C B P D B the subtrees rooted at each of its children are conditionally P(E | D) P(F |independentD) given this node

Remark: We will explain the propagation algorithm on this toy example borrowed from (Pearl, 1988)

Tree Bayesian network

A Each node (except the root) has at most one parent B

C D

E F

P(B | A)

P(C | B) P(D | B)

P(E | D) P(F | D)

Remark: We will explain the propagation algorithm on this toy example borrowed from (Pearl, 1988)

Tree Bayesian network

A Each node (except the root) has at most one parent B Each node separates the tree: its non-descendants and C D the subtrees rooted at each of its children are conditionally E F independent given this node

P(B | A)

P(C | B) P(D | B)

P(E | D) P(F | D)

Tree Bayesian network

Remark: We will explain the propagation algorithm on this toy example borrowed from (Pearl, 1988)

X Top-down propagation 2 P(X) Complexity O(nr ) Y

No evidence

C D

E F

X Top-down propagation 2 P(X) Complexity O(nr ) Y

No evidence

A • P(A): parameter

C D

E F

X Top-down propagation 2 P(X) Complexity O(nr ) Y

No evidence

A • P(A): parameter ∑ B • P(B) = P(B | A)P(A) A C D

E F

X Top-down propagation 2 P(X) Complexity O(nr ) Y

No evidence

A • P(A): parameter P(A) ∑ B • P(B) = P(B | A)P(A) A C D

E F

X Top-down propagation 2 P(X) Complexity O(nr ) Y

No evidence

A • P(A): parameter P(A) ∑ B • P(B) = P(B | A)P(A) A ∑ C D • P(C) = P(C | B)P(B) B

E F

X Top-down propagation 2 P(X) Complexity O(nr ) Y

No evidence

A • P(A): parameter P(A) ∑ B • P(B) = P(B | A)P(A) P(B) A ∑ C D • P(C) = P(C | B)P(B) B

E F

No evidence

A • P(A): parameter P(A) ∑ B • P(B) = P(B | A)P(A) P(B) A ∑ C D • P(C) = P(C | B)P(B) B ∑ E F • P(D) = P(D | B)P(B) B

No evidence

A • P(A): parameter P(A) ∑ B • P(B) = P(B | A)P(A) P(B) P(B) A ∑ C D • P(C) = P(C | B)P(B) B ∑ E F • P(D) = P(D | B)P(B) B

No evidence

A • P(A): parameter P(A) ∑ B • P(B) = P(B | A)P(A) P(B) P(B) A ∑ C D • P(C) = P(C | B)P(B) ( ) ( ) B P D P D ∑ E F • P(D) = P(D | B)P(B) B

No evidence

A • P(A): parameter P(A) ∑ B • P(B) = P(B | A)P(A) P(B) P(B) A ∑ C D • P(C) = P(C | B)P(B) ( ) ( ) B P D P D ∑ E F • P(D) = P(D | B)P(B) B

Top-down propagation Complexity O(nr2)

A • P(A): parameter P(A) ∑ B • P(B) = P(B | A)P(A) P(B) P(B) A ∑ C D • P(C) = P(C | B)P(B) ( ) ( ) B P D P D ∑ E F • P(D) = P(D | B)P(B) B

X Top-down propagation 2 P(X) Complexity O(nr ) Y

• Principle: Propagate the information through the network, starting from the evidence nodes

Three pieces of evidence

A • Evidence: We observe that = = = B C c, E e, and F f

c D

e f

Three pieces of evidence

A • Evidence: We observe that = = = B C c, E e, and F f • Objective: Compute the c D belief BEL(X) = P(X | c,e,f) of each node X

e f

A • Evidence: We observe that = = = B C c, E e, and F f • Objective: Compute the c D belief BEL(X) = P(X | c,e,f) of each node X

e f • Principle: Propagate the information through the network, starting from the X evidence nodes

∝ P(c,e,f | B)P(B)

P(e,f | D)P(D | c) = P(e,f | c) ∝ P(e,f | D)P(D | c)

P(c,e,f | A)P(A) • P(A | c,e,f) = P(c,e,f)

P(c,e,f | B)P(B) • P(B | c,e,f) = P(c,e,f)

P(e,f | D,c)P(D | c) • P(D | c,e,f) = P(e,f | c)

Causal and diagnostic support

A By Bayes’ rule:

c D

e f

∝ P(c,e,f | B)P(B)

P(c,e,f | A)P(A) • P(A | c,e,f) = P(c,e,f)

P(c,e,f | B)P(B) • P(B | c,e,f) = P(c,e,f)

P(e,f | D)P(D | c) = P(e,f | c) ∝ P(e,f | D)P(D | c)

Causal and diagnostic support

A By Bayes’ rule:

c D

e f P(e,f | D,c)P(D | c) • P(D | c,e,f) = P(e,f | c) X

∝ P(c,e,f | B)P(B)

P(c,e,f | A)P(A) • P(A | c,e,f) = P(c,e,f)

P(c,e,f | B)P(B) • P(B | c,e,f) = P(c,e,f)

∝ P(e,f | D)P(D | c)

Causal and diagnostic support

A By Bayes’ rule:

c D

∝ P(c,e,f | B)P(B)

P(c,e,f | A)P(A) • P(A | c,e,f) = P(c,e,f)

P(c,e,f | B)P(B) • P(B | c,e,f) = P(c,e,f)

Causal and diagnostic support

A By Bayes’ rule:

c D

P(c,e,f | A)P(A) • P(A | c,e,f) = P(c,e,f)

∝ P(c,e,f | B)P(B)

Causal and diagnostic support

A By Bayes’ rule:

P(c,e,f | B)P(B) c • P(B | c,e,f) = D P(c,e,f)

P(c,e,f | A)P(A) • P(A | c,e,f) = P(c,e,f)

Causal and diagnostic support

A By Bayes’ rule:

P(c,e,f | B)P(B) c • P(B | c,e,f) = D P(c,e,f) ∝ P(c,e,f | B)P(B)

Causal and diagnostic support

A By Bayes’ rule: P(c,e,f | A)P(A) • P(A | c,e,f) = P(c,e,f) B

P(c,e,f | B)P(B) c • P(B | c,e,f) = D P(c,e,f) ∝ P(c,e,f | B)P(B)

A By Bayes’ rule: P(c,e,f | A)P(A) • P(A | c,e,f) = P(c,e,f) B ∝ P(c,e,f | A)P(A)

P(c,e,f | B)P(B) c • P(B | c,e,f) = D P(c,e,f) ∝ P(c,e,f | B)P(B)

For each X, we compute B ( ¯ ) • evidence ¯ Diagnostic support P below X X c D Bottom-up propagation ( ¯ ) • ¯ evidence Causal support P X above X e f Top-down propagation ( ¯ ) ( ¯ ) BEL(X) ∝ P evidence ¯ X ×P X ¯ evidence X below X above X

• P(e,f | D) = P(e | D)P(f | D) P(c | B) P(e,f | B)

• P(c,e,f | B) = P(c | B)P(e,f | B) P(e | D) P(f | D)

( ¯ ) evidence ¯ Diagnostic support P below X X

A Bottom-up propagation B

c D

e f

P(c | B) P(e,f | B)

• P(c,e,f | B) = P(c | B)P(e,f | B) P(e | D) P(f | D)

( ¯ ) evidence ¯ Diagnostic support P below X X

A Bottom-up propagation B • P(e,f | D) = P(e | D)P(f | D)

c D

e f

P(c | B) P(e,f | B)

• P(c,e,f | B) = P(c | B)P(e,f | B)

( ¯ ) evidence ¯ Diagnostic support P below X X

A Bottom-up propagation B • P(e,f | D) = P(e | D)P(f | D)

c D P(e | D) P(f | D) e f

( ¯ ) evidence ¯ Diagnostic support P below X X

A Bottom-up propagation B • P(e,f | D) = P(e | D)P(f | D)

c D • P(c,e,f | B) = P(c | B)P(e,f | B) P(e | D) P(f | D) e f

( ¯ ) evidence ¯ Diagnostic support P below X X

A Bottom-up propagation B • P(e,f | D) = P(e | D)P(f | D) P(c | B)

c D • P(c,e,f | B) = P(c | B)P(e,f | B) P(e | D) P(f | D) e f

( ¯ ) evidence ¯ Diagnostic support P below X X

A Bottom-up propagation B • P(e,f | D) = P(e | D)P(f | D) P(c | B)

c D • P(c,e,f | B) = P(c | B)P(e,f | B) P(e | D) P(f | D) Compute P(e,f | B): e f ∑ P(e,f | B) = P(e,f | B,D)P(D | B) ∑D X = P(e,f | D)P(D | B) D Y

A Bottom-up propagation B • P(e,f | D) = P(e | D)P(f | D) P(c | B) P(e,f | B)

c D • P(c,e,f | B) = P(c | B)P(e,f | B) P(e | D) P(f | D) Compute P(e,f | B): e f ∑ P(e,f | B) = P(e,f | B,D)P(D | B) ∑D X = P(e,f | D)P(D | B) D Y

(Bottom-up¯ ) evidence ¯ P below Y X

( ¯ ) evidence ¯ Diagnostic support P below X X

( ¯ ) evidence ¯ Diagnostic support P below X X

→ BEL(B) ∝ P(c,e,f | B)P(B)

P(A)

• P(A): parameter

∑ • P(B) = P(B | A)P(A) A

( ¯ ) ¯ evidence Causal support P X above X

c D

e f

X (Bottom-up¯ ) evidence ¯ P below Y X Y

P(A)

→ BEL(A) ∝ P(c,e,f | A)P(A) ∑ • P(B) = P(B | A)P(A) A

( ¯ ) ¯ evidence Causal support P X above X

B • P(A): parameter

c D

e f

X (Bottom-up¯ ) evidence ¯ P below Y X Y

P(A)

∑ • P(B) = P(B | A)P(A) A

( ¯ ) ¯ evidence Causal support P X above X

B • P(A): parameter → ∝ | c D BEL(A) P(c,e,f A)P(A)

e f

X (Bottom-up¯ ) evidence ¯ P below Y X Y

→ BEL(B) ∝ P(c,e,f | B)P(B)

( ¯ ) ¯ evidence Causal support P X above X

B • P(A): parameter → ∝ | c D BEL(A) P(c,e,f A)P(A) ∑ • P(B) = P(B | A)P(A) e f A

X (Bottom-up¯ ) evidence ¯ P below Y X Y

( ¯ ) ¯ evidence Causal support P X above X

A P(A) B • P(A): parameter → ∝ | c D BEL(A) P(c,e,f A)P(A) ∑ • P(B) = P(B | A)P(A) e f A

X (Bottom-up¯ ) evidence ¯ P below Y X Y

A P(A) B • P(A): parameter → ∝ | c D BEL(A) P(c,e,f A)P(A) ∑ • P(B) = P(B | A)P(A) e f A → BEL(B) ∝ P(c,e,f | B)P(B)

X (Bottom-up¯ ) evidence ¯ P below Y X Y

→ BEL(D) ∝ P(e,f | D)P(D | c)

∑ • P(D | c) = P(D | B,c)P(B | c) B

P(B | c)

Top-down( ¯ ) ¯ evidence P X above Y

( ¯ ) ¯ evidence Causal support P X above X

A P(A) B

c D

e f

X (Bottom-up¯ ) evidence ¯ P below Y X Y

( ¯ ) ¯ evidence Causal support P X above X

A ∑ • P(D | c) = P(D | B,c)P(B | c) P(A) B B

c D

e f

X (Bottom-up¯ ) evidence ¯ P below Y X Y

( ¯ ) ¯ evidence Causal support P X above X

A ∑ • P(D | c) = P(D | B,c)P(B | c) P(A) ∑B B = P(D | B)P(B | c) B

c D

e f

X (Bottom-up¯ ) evidence ¯ P below Y X Y

( ¯ ) ¯ evidence Causal support P X above X

A ∑ • P(D | c) = P(D | B,c)P(B | c) P(A) ∑B B = P(D | B)P(B | c) B P(B | c) c D

e f

X (Bottom-up¯ ) evidence ¯ P below Y X Y

( ¯ ) ¯ evidence Causal support P X above X

( ¯ ) ¯ evidence Causal support P X above X

In general • Use a topological ordering • Complexity: O(rd↓ + r2 + r)

Summary

A Algorithm

c D

e f

X Top-down (Bottom-up¯ ) ( ¯ ) evidence ¯ ¯ evidence P below Y X P X above Y Y

In general • Use a topological ordering • Complexity: O(rd↓ + r2 + r)

Summary

A Algorithm ( ¯ ) • evidence ¯ Diagnostic support P below X X B Bottom-up propagation

c D

e f

X Top-down (Bottom-up¯ ) ( ¯ ) evidence ¯ ¯ evidence P below Y X P X above Y Y

Summary

A Algorithm ( ¯ ) • evidence ¯ Diagnostic support P below X X B Bottom-up propagation ( ¯ ) • ¯ evidence Causal support P X above X c D Top-down propagation

e f

X Top-down (Bottom-up¯ ) ( ¯ ) evidence ¯ ¯ evidence P below Y X P X above Y Y

Summary

A Algorithm ( ¯ ) • evidence ¯ Diagnostic support P below X X B Bottom-up propagation ( ¯ ) • ¯ evidence Causal support P X above X c D Top-down propagation

e f In general

X Top-down (Bottom-up¯ ) ( ¯ ) evidence ¯ ¯ evidence P below Y X P X above Y Y

Summary

A Algorithm ( ¯ ) • evidence ¯ Diagnostic support P below X X B Bottom-up propagation ( ¯ ) • ¯ evidence Causal support P X above X c D Top-down propagation

e f In general • Use a topological ordering

X Top-down (Bottom-up¯ ) ( ¯ ) evidence ¯ ¯ evidence P below Y X P X above Y Y

A Algorithm ( ¯ ) • evidence ¯ Diagnostic support P below X X B Bottom-up propagation ( ¯ ) • ¯ evidence Causal support P X above X c D Top-down propagation

e f In general • Use a topological ordering • Complexity: O(rd↓ + r2 + r) X Top-down (Bottom-up¯ ) ( ¯ ) evidence ¯ ¯ evidence P below Y X P X above Y Y

• Asynchronous / parallel updates: acknowledgements (Pearl, 1982, 1986, 1988)

Additional remarks

A • If the evidence node is not a leaf: add a phantom node B

c D

e f

X Top-down (Bottom-up¯ ) ( ¯ ) evidence ¯ ¯ evidence P below Y X P X above Y Y

Additional remarks

A • If the evidence node is not a leaf: add a phantom node

B • The calculations can be written as matrix products c D − Belief, causal and diagnostic supports, messages ∼ Vectors e f − CPT ∼ Matrix

X Top-down (Bottom-up¯ ) ( ¯ ) evidence ¯ ¯ evidence P below Y X P X above Y Y

A • If the evidence node is not a leaf: add a phantom node

B • The calculations can be written as matrix products c D − Belief, causal and diagnostic supports, messages ∼ Vectors e f − CPT ∼ Matrix

• Asynchronous / parallel X Top-down (Bottom-up¯ ) ( ¯ ) updates: acknowledgements evidence ¯ ¯ evidence P below Y X P X above Y (Pearl, 1982, 1986, 1988) Y

Reminders on probability theory

Bayesian networks

Belief propagation in trees

Belief propagation in polytrees

37/40 © 2018 Nokia Public Separation properties • Given a node, the non- descendants and the subtrees rooted at each child are independent • If we don’t condition on a node nor any of its descendants, the inversed subtrees rooted at its ancestors are independent

Polytree (or singly-connected) Bayesian network

The underlying undirected graph is a tree

D I

G B

38/40 © 2018 Nokia Public • Given a node, the non- descendants and the subtrees rooted at each child are independent • If we don’t condition on a node nor any of its descendants, the inversed subtrees rooted at its ancestors are independent

Polytree (or singly-connected) Bayesian network

The underlying undirected graph is a tree

Separation properties D I

G B

Polytree (or singly-connected) Bayesian network

The underlying undirected graph is a tree

Separation properties D I • Given a node, the non- descendants and the G B subtrees rooted at each child are independent

The underlying undirected graph is a tree

Separation properties D I • Given a node, the non- descendants and the G B subtrees rooted at each child are independent • If we don’t condition on a L node nor any of its descendants, the inversed subtrees rooted at its ancestors are independent

( ) ↑ AX ↑ BX ¯ • evidence ¯ Diagnostic support P below X X Bottom-up propagation ↓ XC and ↓ XD are independent given X ( ¯ ) • ¯ evidence Causal support P X above X Top-down propagation ↑ AX and ↑ BX are independent ↓ XC ↓ XD

Belief propagation

A B

C D

39/40 © 2018 Nokia Public ( ) ↑ AX ↑ BX ¯ • evidence ¯ Diagnostic support P below X X Bottom-up propagation ↓ XC and ↓ XD are independent given X ( ¯ ) • ¯ evidence Causal support P X above X Top-down propagation ↑ AX and ↑ BX are independent ↓ XC ↓ XD

Belief propagation

( ¯ ) ( ¯ ) | ∝ evidence ¯ × ¯ evidence P(X evidence) P below X X P X above X

A B

C D

39/40 © 2018 Nokia Public ( ) ↑ BX ¯ • evidence ¯ Diagnostic support P below X X Bottom-up propagation ↓ XC and ↓ XD are independent given X ( ¯ ) • ¯ evidence Causal support P X above X Top-down propagation ↑ AX and ↑ BX are independent ↓ XC ↓ XD

Belief propagation

( ¯ ) ( ¯ ) | ∝ evidence ¯ × ¯ evidence P(X evidence) P below X X P X above X

↑ AX

A B

C D

39/40 © 2018 Nokia Public ( ¯ ) • evidence ¯ Diagnostic support P below X X Bottom-up propagation ↓ XC and ↓ XD are independent given X ( ¯ ) • ¯ evidence Causal support P X above X Top-down propagation ↑ AX and ↑ BX are independent ↓ XC ↓ XD

Belief propagation

( ¯ ) ( ¯ ) | ∝ evidence ¯ × ¯ evidence P(X evidence) P below X X P X above X

↑ AX ↑ BX

A B

C D

Belief propagation

( ¯ ) ( ¯ ) | ∝ evidence ¯ × ¯ evidence P(X evidence) P below X X P X above X

↑ AX ↑ BX

A B

C D

↓ XC

Belief propagation

( ¯ ) ( ¯ ) | ∝ evidence ¯ × ¯ evidence P(X evidence) P below X X P X above X

↑ AX ↑ BX

A B

C D

↓ XC ↓ XD

Belief propagation

( ¯ ) ( ¯ ) | ∝ evidence ¯ × ¯ evidence P(X evidence) P below X X P X above X

( ) ↑ AX ↑ BX ¯ • evidence ¯ Diagnostic support P below X X A B Bottom-up propagation ↓ XC and ↓ XD are independent given X X

C D

↓ XC ↓ XD

( ¯ ) ( ¯ ) | ∝ evidence ¯ × ¯ evidence P(X evidence) P below X X P X above X

( ) ↑ AX ↑ BX ¯ • evidence ¯ Diagnostic support P below X X A B Bottom-up propagation ↓ XC and ↓ XD are independent given X X ( ¯ ) • ¯ evidence Causal support P X above X C D Top-down propagation ↑ AX and ↑ BX are independent ↓ XC ↓ XD

• Belief propagation A time-efficient algorithm for computing the belief − Asynchronous, parallelizable − Exact in (poly)trees − In general, extended to the junction tree algorithm and to other (approximate) algorithms

Conclusion

40/40 © 2018 Nokia Public • Belief propagation A time-efficient algorithm for computing the belief − Asynchronous, parallelizable − Exact in (poly)trees − In general, extended to the junction tree algorithm and to other (approximate) algorithms

Conclusion

• Bayesian networks A memory-efficient way of storing a PMF by leveraging conditional independencies between variables