bayesian network

Bayesian Network

By Zhang Liliang

Key Point Today

• Intro to Bayesian Network• Usage of Bayesian Network• Reasoning BN: D-separation

Bayesian Network Definition

Difficulty {easy, hard}Intelligence {low, high}Grade {A, B, C}SAT {low_mark, high_mark}Letter {No, Yes}

Bayes networks defines Joint Distribution in term of a graph over a collection of random variable.

P(D, I, G, S, L)= ?

Joint Distribution of BN

General Form:P(D, I, G, S, L) = P(L|D, I, G, S) *P(S|D, I, G) *P(G|D, I) *P(I|D) *P(D)

D

G

L

S

I

Can the formula be simplify?

Conditional IndependenceFor (sets of) random variables X,Y,ZX is conditional independence of Y given Z, Denotes as P (X ⊥ Y | Z), if:• – P(X, Y|Z) = P(X|Z) P(Y|Z)• – P(X|Y,Z) = P(X|Z)• – P(Y|X,Z) = P(Y|Z)

Bayesian Network Definition(2)A Bayesian network is a directed acyclic graph(DAG) and a set of conditional probability distribution(CPD).

P(D, I, G, S, L) = P(D) * P(I) * P(G|D, I) * P(S|I) * P(L|G)

Usages of BN: Reasoning

3 kinds of reasoning:• Causal Reasoning• Evidential Reasoning• Intercausal Reasoning

D

G

L

S

I

Causal Reasoning

Evidential Reasoning

Intercausal Reasoning

How to reasoning?

• For certain cases, tractable - Full observed set of variable - just one variable unobserved• In general, intractable…(NP-complete)

How to deal with the problem?An intuitive solution: D-separation

Conditional Independence: RevisitedFor (sets of) random variables X,Y,ZX is conditional independence of Y given Z, Denotes as P (X ⊥ Y | Z), if:• – P(X, Y|Z) = P(X|Z) P(Y|Z)• – P(X|Y,Z) = P(X|Z)• – P(Y|X,Z) = P(X|Z)

D

G

L

S

I

Given an observation of G, is L is conditional independence of D?Given an observation of I, is G conditional independence of S ? Given an observation of G, is D conditional independence of I ?

A method may simplify the calculation when reasoning : to find out more variables which satisfied with conditional independence.

Three Easy Network about Conditional Independence

Tail to Tail Head to HeadHead to Tail

Head to Tail

(D L ) ? ⊥

(D L| G) ?⊥

D

G

L

S

INo

Yes

Tail to Tail

(G S) ?⊥

(G S| I) ?⊥

D

G

L

S

INo

Yes

Head to Head

(D I)?⊥

(D I| G) ?⊥

D

G

L

S

IYes

No

X and Y are conditionally independent given Z, if and only if X and Y are D-separated by Z

Suppose we have three sets of random variables: X, Y and Z

X and Y are D-separated by Z (and therefore conditionally independence, given Z) iff every path from any variable in X to any variable in Y is blocked

A path from variable A to variable B is blocked if it includes a node such that either1.arrows on the path meet either head-to-tail or tail-to-tail at the node and this node is in Z2.the arrows meet head-to-head at the node, and neither the node, nor any of its descendants, is in Z

Summary

• Bayesian Network = Directed Acyclic Graph + Conditional Probability Distribution

• Joint Distribution:• Three type of Reasoning BN: causal,

evidential, intercausal• Conditional Independence & D-separation

Reference

• Machine Learning, CMU• Probabilistic Graphical Models, Stanford, on

Coursera.• SamIam: a comprehensive tool, UCLA

Thanks

bayesian network

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