Bayesian Networks: Belief Propagation in Singly Connected Networks

Transcription

1 Bayesian Networks: Belief Propagation in Singly Connected Networks Huizhen u janey.yu@cs.helsinki.fi Dept. Computer Science, Univ. of Helsinki Probabilistic Models, Spring, 2010 Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

2 Outline Belief Propagation In Chains In Trees Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

3 Form of Evidence and Notation We denote evidence (a finding) of X = {X v, v V } by e. Formally, we think of e as a function of x taking values in {0, 1}, representing a statement that some elements of x are impossible, i.e., {x e(x) = 1} is the set of possible values of x based on the evidence e. We also refer to this event as e. We consider e that can be written in the factor form e(x) = v V l v (x v ), where l v (x v ) {0, 1}. For A V, we use e A to denote the partial evidence of X A : e A (x A ) = v A l v (x v ). Other short-hand notation we will use: p(x A & e) = P(X A = x A, e), p(x A & e A x B ) = P(X A = x A, e A X B = x B ) = P(X A = x A X B = x B ) e A (x A ), and p(x A e) denotes the conditional PMF of X A given the event e. Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

4 Motivation Inference tasks we consider here: calculate p(x v e), x v and P(e) for P that is directed Markov w.r.t. a DAG G. Note that if we know P(e), then we can calculate the posterior probability of a single x given e easily: p(x e) = p(x & e)/p(e) = p(x v x pa(v) ) l v (x v ) /P(e). Since P(X = x, e) = p(x) e(x), in principle we can calculate P(X v = x v, e) = X P(X V \{v} = x V \{v}, X v = x v, e), x V \{v} v V P(e) = X x P(X = x, e). But such calculation is not easy in most problems when V is large. The function p(x v e) is referred to as the belief of x v. Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

5 Features of the Algorithms to be Introduced In the algorithms to be introduced, the DAG G is treated also as the architecture for distributed computation: Nodes: associated with autonomous processors Edges: communication links between processors The independence relations represented by the DAG are exploited to separate the total evidence into pieces and streamline the computation. The algorithms have performance guarantee on DAGs with simple structures G has no loops. But they have also been used successfully as approximate inference algorithms on loopy graphs. Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

6 In Chains Outline Belief Propagation In Chains In Trees Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

7 In Chains Evidence Structure in a Chain Suppose G is a chain. Consider a vertex v with parent u and child w: e u + u v w e v We write e as three pieces of evidence, e = (e u +, e v, e v ), where e u +: partial evidence of X an(v) e v : partial evidence of X v e v : partial evidence of X de(v) We want to compute p(x v & e) = P(X v = x v, e) for all x v. Since we have P(X an(v), X v, X de(v) ) = P(X an(v) ) P(X v X u) P(X de(v) X v ), p`(x u, x v ) & e) = p(x u & e u +) p(x v & e v x u) p(e v x v ). If v can get the first and third terms from u and w respectively, then v can calculate its marginal p(x v & e) by summing over x u. Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

8 In Chains If node v receives Message Passing in a Chain from parent u the probabilities of x u and partial evidence e u + on u s side: π u,v (x u) = p(x u & e u +), x u; from child w the likelihoods of x v based on the partial evidence e v on w s side: λ w,v (x v ) = p(e v x v ), x v, then node v can calculate p(x v & e) = X x u π u,v (x u) p(x v x u) l v (x v ) λ w,v (x v ), x v. What u and w need from v in order to calculate their marginal probabilities? Parent u needs for all x u, the likelihood of x u based on e u = (e v, e v ): λ v,u(x u) = p(e u x u) = X x v p(x v & e v x u) p(e v x v ) = X x v p(x v x u) l v (x v ) λ w,v (x v ). Child w needs for all x v, the probability of x v and e v + = (e u +, e v ): π v,w (x v ) = p(x v & e v +) = X x u π u,v (x u) p(x v x u) l v (x v ). Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

9 In Chains Algorithm Summary λ-messages (likelihoods) π-messages (probabilities) Each node v when receiving the message λ w,v from its child, sends to its parent u λ v,u(x u) = X x v p(x v x u) l v (x v ) λ w,v (x v ), x u; when receiving the message π u,v from its parent, sends to its child w π v,w (x v ) = X x u π u,v (x u) p(x v x u) l v (x v ), x v ; when receiving both messages, calculates p(x v & e) = X x u π u,v (x u) p(x v x u) l v (x v ) λ w,v (x v ), x v, P(e) = X x v p(x v & e), p(x v e) = p(x v & e)/p(e). Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

10 In Trees Outline Belief Propagation In Chains In Trees Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

11 In Trees Evidence Structure in a Rooted Tree Suppose G is a rooted tree. Then G m = G. Consider a vertex v with parent u and children w 1,..., w m: We write the total evidence e as several pieces of evidence, e nd(v) where e = (e nd(v), e v, e Tw1,..., e Twm ), e nd(v) : partial evidence of X nd(v) u v e v : partial evidence of X v e Tw, w ch(v): partial evidence of the variables associated with the subtree T w rooted at w, i.e., X {w} de(w) e Tw1 w 1 w m Since P(X nd(v), X v, X de(v) ) = P(X nd(v) ) P(X v X u) P(X Tw X v ), p`(x u, x v ) & e = p(x u & e nd(v) ) p(x v & e v x u) Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25 p(e Tw x v ).

12 In Trees Message Passing in a Rooted Tree From p`(x u, x v ) & e = p(x u & e nd(v) ) p(x v & e v x u) p(e Tw x v ). we see that if v receives from parent u the probabilities of x u and evidence e nd(v) for all x u: e nd(v) u π u,v (x u) = p(x u & e nd(v) ), x u; from every child w the likelihoods of all x v based on the evidence e Tw : λ w,v (x v ) = p(e Tw x v ), x v, w 1 v w m then node v can calculate p(x v & e) = X π u,v (x u) p(x v x u) l v (x v ) λ w,v (x v ). x u e Tw1 Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

13 In Trees Message Passing in a Rooted Tree What do nodes u and w need from v in order to calculate their marginals? Parent u needs the likelihoods of x u based on e Tv for all x u: λ v,u(x u) = X p(x v & e v x u) p(e Tw x v ) x v = X x v p(x v x u) l v (x v ) Child w needs for all x v, the probability of x v and «e nd(w) = e nd(v), e Tw : w ch(v)\{w} λ w,v (x v ). end(v) u v π v,w (x v ) = p(x v & e nd(w) ) X = p(x u & e nd(v) ) p(x v & e v x u) x u p(e Tw x v ) w ch(v)\{w} X = π u,v (x u) p(x v x u) l v (x v ) x u λ w,v (x v ). w ch(v)\{w} w1 etw 1 wm Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

14 In Trees Each node v sends to its parent u Algorithm Summary λ v,u(x u) = X x v p(x v x u) l v (x v ) sends to its child w X π v,w (x v ) = π u,v (x u) p(x v x u) l v (x v ) x u when receiving all messages, calculates X p(x v & e) = π u,v (x u) p(x v x u) l v (x v ) x u λ w,v (x v ), x u; w ch(v)\{w} P(e) = X x v p(x v & e), p(x v e) = p(x v & e)/p(e). λ w,v (x v ), x v ; λ w,v (x v ) x v, Message passing schemes: (i) Each node can send a message to a linked node if it has received messages from all the other linked nodes. (ii) Each node can send updated messages to linked nodes whenever it gets a new message from some node. Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

15 In Trees Illustration of Parallel Updating From J. Peal s book, 1988: At time 0, each node of the tree has calculated its own marginal. At time 1, two new pieces of evidence arrive and trigger new messages. After time 5, all nodes have updated their marginals incorporating the new evidence. λ-message: Data Data π-message: t = 0 t = 1 t = 2 t = 5 t = 4 t = 3 Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

16 Outline Belief Propagation In Chains In Trees Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

17 Definition of a Singly Connected Network Definition: a DAG G is singly connected, if its undirected version G is a tree. Such a G is also called a polytree. In a polytree G: Each node can have multiple parents and children. But there is only one trail between each pair of nodes. Consider a vertex v with parents u 1,... u n and children w 1,..., w m. When v is viewed as the center, the branch of the polytree containing one of its parents or children is a sub-polytree. Denote T vui, i = 1..., n: the sub-polytree containing the node u i, resulting from removing the edge (u i, v); T vwi, i = 1,..., m: the sub-polytree containing the node w i, resulting from removing the edge (v, w i ). Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

18 Evidence Structure in a Singly Connected Network For a sub-polytree T, denote X T : the variables associated with nodes in T etvu 1 u1 un e T : the partial evidence of X T v We have w1 wm P(X Tvu1,..., X Tvun ) = P(X Tvu1 ) P(X Tvun ), and etvw 1 P(X Tvw1,..., X Tvwm X v ) = P(X Tvw1 X v ) P(X Tvwm X v ). (Why? We may argue this using (DG) or d-separation the latter is also simple in this case because there is only one trail between each pair of nodes.) Therefore, p `xpa(v), x v & e = u pa(v) p(x u & e Tvu ) p`x v & e v x pa(v) p(e Tvw x v ). Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

19 From p `xpa(v), x v & e Message Passing in a Singly Connected Network = u pa(v) p(x u & e Tvu ) p`x v & e v x pa(v) p(e Tvw x v ). we see that v can calculate its marginal if it receives messages π u,v from all parents, where etvu 1 u1 un π u,v (x u) = p(x u & e Tvu ), x u; v and λ w,v from all children, where λ w,v (x v ) = p(e Tvw x v ), x v. w1 wm etvw 1 Then, p(x v & e) is given by X p(x v & e) = x pa(v) u pa(v) π u,v (x u) p(x v x pa(v) ) l v (x v ) λ w,v (x v ) Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

20 Message Passing in a Singly Connected Network What do parents need from v in order to calculate their marginals? A parent u needs the likelihoods of all x u based on the partial evidence e Tuv from the sub-polytree on v s side with respect to u: λ v,u(x u) = X x v X x pa(v)\{u} p`x v & e v x pa(v) = X x v X x pa(v)\{u} p`x v x pa(v) lv (x v ) u pa(v)\{u} u pa(v)\{u} p(x u & e Tvu ) p(e Tvw x v ) π u,v (x u ) λ w,v (x v ). e Tvu1 u 1 u n v w 1 w m e Tvw1 Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

21 Message Passing in a Singly Connected Network What do children need from v in order to calculate their marginals? A child w needs for all x v, the probability of x v and the partial evidence e Twv from the sub-polytree on v s side with respect to w: p(x v & e Twv ) = X p`x v & e v x pa(v) p`x u & e Tvu p(e Tvw xv ) x pa(v) u pa(v) w ch(v)\{w} X = p`x v x pa(v) lv (x v ) π u,v (x u) λ w,v (x v ). x pa(v) u pa(v) w ch(v)\{w} e Tvu1 u 1 u n v w 1 w m e Tvw1 Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

22 Algorithm Summary Each node v sends to each u of its parents λ v,u(x u) = X X p(x v x pa(v) ) l v (x v ) x v x pa(v)\{u} π u,v (x u ) λ w,v (x v ), x u; u pa(v)\{u} sends to each w of its children π v,w (x v ) = λ w,v (x v ) X p(x v x pa(v) ) l v (x v ) x pa(v) w ch(v)\{w} π u,v (x u), x v ; u pa(v) when receiving all messages from parents and children, calculates p(x v & e) = λ w,v (x v ) X π u,v (x u) p(x v x pa(v) ) l v (x v ), x v, x pa(v) u pa(v) P(e) = X x v p(x v & e), p(x v e) = p(x v & e)/p(e). Message passing schemes: (i) Each node can send a message to a linked node if it has received messages from all the other linked nodes. (ii) Each node can send updated messages to linked nodes whenever it gets a new message from some node. Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

23 Example of Noisy-Or Gate x i, y i, y {0, 1}. P(X i = 1) = p i, P( i = 1 X i = 0) = 0, P( i = 1 X i = 1) = 1 q i. p(y x 1,..., x n) ( Q i:xi =1 q i, if y = 0; = 1 Q i:x i =1 q i, if y = 1. X X X 1 2 n 1 2 n = OR( 1, 2,... n ) Express the message π Xi, i (x i ) in the vector form ˆ π Xi, i (1), π Xi, i (0) : π Xi, i = ˆ p i, 1 p i. Similarly, express π i, (y i ) as ˆ π i, (1), π i, (0) : π i, (y i ) = X π Xi, i (x i )p(y i x i ), so π i, = ˆ p i (1 q i ), p i q i +(1 p i ). x i {0,1} Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

24 Example of Noisy-Or Gate Suppose e : { = 1} is received. Then, sends a message λ,i = ˆ λ,i (1), λ,i (0) to each i, where λ,i (y i ) = X X p(1 y 1,..., y n) π j, (y j ). k i y k {0,1} j i (What are these values?) Subsequently, each i sends to X i the message λ i,x i (x i ): λ i,x i (1) = (1 q i ) λ,i (1) + q i λ,i (0), λ i,x i (0) = λ,i (0). Each X i can calculate its marginal and posterior probability of X i = 1 as P(X i = 1, e) = P(X i = 1) λ i,x i (1), P(X i = 0, e) = P(X i = 0) λ i,x i (0), P(X i = 1 e) = p i λ i,x i (1) p i λ i,x i (1) + (1 p i ) λ i,x i (0). Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25

25 Generalizations and Further Reading Find most probable configurations: max x p(x & e) Conditioning: G is not a tree. We condition on certain variables to create several singly connected networks and then fuse together the calculated results. Loopy belief propagation: G is not a tree, but we apply the message passing algorithm any way. Algorithm variants and convergence analysis are active research topics. Further reading: 1. Judea Pearl. Probabilistic Reasoning in Intelligent Systems, Morgan Kaufmann, Chap. 4. Huizhen u (U.H.) Bayesian Networks: Belief Propagation in Singly Connected Networks Feb / 25