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 Independence For (sets of) random variables X,Y,Z X 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)

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 In BN, it can be simplify as: P(D, I, G, S, L) = P(D) * P(I) * P(G|D, I) * P(S|I) * P(L|G) Parameters: 2*2*3*2*2-1=47 Parameters: = 15

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: Revisited For (sets of) random variables X,Y,Z X 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 Head Head to Tail

(D ⊥ L ) ? (D ⊥ L| G) ? D G L S I No Yes

Tail to Tail (G ⊥ S) ? (G ⊥ S| I) ? D G L S I No Yes

Head to Head (D ⊥ I)? (D ⊥ I| G) ? D G L S I Yes 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 either 1.arrows on the path meet either head-to-tail or tail-to-tail at the node and this node is in Z 2.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