Outline. Bayesian Networks: Belief Propagation in Singly Connected Networks. Form of Evidence and Notation. Motivation

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1 Outline Bayesian : in Singly Connected Huizhen u Dept. Computer Science, Uni. of Helsinki Huizhen u (U.H.) Bayesian : in Singly Connected Feb / 25 Huizhen u (U.H.) Bayesian : in Singly Connected Feb / 25 Form of Eidence and Notation We denote eidence (a finding) of X = {X, V } by e. Formally, we think of e as a function of x taking alues in {0, 1}, representing a statement that some elements of x are impossible, i.e., {x e(x) = 1} is the set of possible alues of x based on the eidence e. We also refer to this eent as e. We consider e that can be written in the factor form e(x) = V l (x ), where l (x ) {0, 1}. For A V, we use e A to denote the partial eidence of X A : e A (x A ) = A l (x ). 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 ), Motiation nce tasks we consider here: calculate p(x e), x and P(e) for P that is directed Marko w.r.t. a DAG G. Note that if we know P(e), then we can calculate the posterior probability of a single x gien e easily: p(x e) = p(x & e)/p(e) = p(x x pa() ) l (x ) /P(e). Since P(X = x, e) = p(x) e(x), in principle we can calculate P(X = x, e) = X P(X V \{} = x V \{}, X = x, e), x V \{} P(e) = X x V P(X = x, e). But such calculation is not easy in most problems when V is large. The function p(x e) is referred to as the belief of x. and p(x A e) denotes the conditional PMF of X A gien the eent e. Huizhen u (U.H.) Bayesian : in Singly Connected Feb / 25 Huizhen u (U.H.) Bayesian : in Singly Connected Feb / 25

2 Features of the Algorithms to be Introduced Outline 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 eidence into pieces and streamline the computation. The algorithms hae performance guarantee on DAGs with simple structures G has no loops. But they hae also been used successfully as approximate inference algorithms on loopy graphs. Huizhen u (U.H.) Bayesian : in Singly Connected Feb / 25 Huizhen u (U.H.) Bayesian : in Singly Connected Feb / 25 Eidence Structure in a Chain Suppose G is a chain. Consider a ertex with parent u and child w: e u + u w e We write e as three pieces of eidence, e = (e u +, e, e ), where e u +: partial eidence of X an() e : partial eidence of X e : partial eidence of X de() We want to compute p(x & e) = P(X = x, e) for all x. Since we hae P(X an(), X, X de() ) = P(X an() ) P(X X u) P(X de() X ), p`(, x ) & e) = p( & e u +) p(x & e ) p(e x ). If can get the first and third terms from u and w respectiely, then can calculate its marginal p(x & e) by summing oer. Huizhen u (U.H.) Bayesian : in Singly Connected Feb / 25 If node receies Message Passing in a Chain from parent u the probabilities of and partial eidence e u + on u s side: π u, () = p( & e u +), ; from child w the likelihoods of x based on the partial eidence e on w s side: λ w, (x ) = p(e x ), x, then node can calculate p(x & e) = X π u, () p(x ) l (x ) λ w, (x ), x. What u and w need from in order to calculate their marginal probabilities? Parent u needs for all, the likelihood of based on e u = (e, e ): λ,u() = p(e u ) = X x p(x & e ) p(e x ) = X x p(x ) l (x ) λ w, (x ). Child w needs for all x, the probability of x and e + = (e u +, e ): π,w (x ) = p(x & e +) = X π u, () p(x ) l (x ). Huizhen u (U.H.) Bayesian : in Singly Connected Feb / 25

3 Algorithm Summary Outline λ-messages (likelihoods) π-messages (probabilities) Each node when receiing the message λ w, from its child, sends to its parent u λ,u() = X x p(x ) l (x ) λ w, (x ), ; when receiing the message π u, from its parent, sends to its child w π,w (x ) = X π u, () p(x ) l (x ), x ; when receiing both messages, calculates p(x & e) = X π u, () p(x ) l (x ) λ w, (x ), x, P(e) = X x p(x & e), p(x e) = p(x & e)/p(e). Huizhen u (U.H.) Bayesian : in Singly Connected Feb / 25 Eidence Structure in a Rooted Tree Suppose G is a rooted tree. Then G m = G. Consider a ertex with parent u and children w 1,..., w m: We write the total eidence e as seeral pieces of eidence, where e = (e nd(), e, e Tw1,..., e Twm ), e nd() : partial eidence of X nd() e : partial eidence of X e Tw, w ch(): partial eidence of the ariables associated with the subtree T w rooted at w, i.e., X {w} de(w) Since e nd() e Tw1 P(X nd(), X, X de() ) = P(X nd() ) P(X X u) p`(, x ) & e = p( & e nd() ) p(x & e ) w 1 u w m P(X Tw X ), p(e Tw x ). Huizhen u (U.H.) Bayesian : in Singly Connected Feb / 25 Huizhen u (U.H.) Bayesian : in Singly Connected Feb / 25 From Message Passing in a Rooted Tree p`(, x ) & e = p( & e nd() ) p(x & e ) we see that if receies Bayesian : Message Passing in Singly Connected from parent u the probabilities of and eidence e nd() for all : π u, () = p( & e nd() ), ; from eery child w the likelihoods of all x based on the eidence e Tw : λ w, (x ) = p(e Tw x ), x, then node can calculate Huizhen u Dept. Computer Science, Uni. of Helsinki e nd() w 1 p(e Tw x ). Huizhen u (U.H.) Bayesian : Message Passing in Singly Connected Feb / 20 Form of eidence, pieces of eidence, e Tu1 u n e Tw1 w m e nd() e Tw1 λ π p(x & e) = X π u, () p(x ) l (x ) u λ w, (x ). Huizhen u (U.H.) Bayesian : in Singly Connected Feb / 25 w m Lo Co Huizhen Lo Co

4 Message Passing in a Rooted Tree What do nodes u and w need from in order to calculate their marginals? Parent u needs the likelihoods of based on e T for all : λ,u() = X p(x & e ) p(e Tw x ) x = X x p(x ) l (x ) Child w needs for all x, the probability of x and «e nd(w) = e nd(), e Tw : λ w, (x ). π,w (x ) = p(x & e nd(w) ) = p( & e nd() ) p(x & e ) p(e Tw x ) = π u, () p(x ) l (x ) end() etw 1 w1 λ w, (x ). u wm Each node sends to its parent u Algorithm Summary λ,u() = X x p(x ) l (x ) sends to its child w π,w (x ) = π u, () p(x ) l (x ) when receiing all messages, calculates p(x & e) = π u, () p(x ) l (x ) λ w, (x ), ; P(e) = X x p(x & e), p(x e) = p(x & e)/p(e). λ w, (x ), x ; λ w, (x ) x, Message passing schemes: (i) Each node can send a message to a linked node if it has receied messages from all the other linked nodes. (ii) Each node can send updated messages to linked nodes wheneer it gets a new message from some node. Huizhen u (U.H.) Bayesian : in Singly Connected Feb / 25 Huizhen u (U.H.) Bayesian : in Singly Connected Feb / 25 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 eidence arrie and trigger new messages. After time 5, all nodes hae updated their marginals incorporating the new eidence. Outline Data Data t = 0 t = 1 t = 2 λ-message: π-message: t = 5 t = 4 t = 3 Huizhen u (U.H.) Bayesian : in Singly Connected Feb / 25 Huizhen u (U.H.) Bayesian : in Singly Connected Feb / 25

5 Definition of a Singly Connected Network Eidence Structure in a Singly Connected Network Definition: a DAG G is singly connected, if its undirected ersion G is a tree. Such a G is also called a polytree. For a sub-polytree T, denote X T : the ariables associated with nodes in T etu 1 u1 un In a polytree G: e T : the partial eidence of X T Each node can hae multiple parents and children. We hae w1 wm But there is only one trail between each pair of nodes. P(X Tu1,..., X Tun ) = P(X Tu1 ) P(X Tun ), and etw 1 Consider a ertex with parents u 1,... u n and children w 1,..., w m. When is iewed as the center, the branch of the polytree containing one of its parents or children is a sub-polytree. Denote T ui, i = 1..., n: the sub-polytree containing the node u i, resulting from remoing the edge (u i, ); Huizhen u T wi, i = 1,..., m: the sub-polytree containing the node w i, resulting Dept. Computer Science, Uni. of Helsinki from remoing the edge (, w i). Huizhen u (U.H.) Bayesian : Bayesian : inmessage Singly Connected Passing in Singly ConnectedFeb / 25 From p `xpa(), x & e Huizhen u Dept. Computer Science, Uni. of Helsinki Message Passing in a Singly Connected Network = u pa() we see that can calculate its marginal if it receies messages π u, from all parents, where π u, () = p( & e Tu ), ; and λ w, from all children, where λ w, (x ) = p(e Tw x ), x. Then, p(x & e) is gien by p(x & e) = x pa() u pa() Bayesian : Message Passing in Singly Connected Huizhen u (U.H.) Bayesian : Message Passing in Singly Connected Feb / 17 p( & e Tu ) p`x & e x pa() Form of eidence w1 wm p(e Tw x ). Huizhen u (U.H.) Bayesian : Message Passing in Singly Connected Feb / 17 Form of eidence etu 1 un w1 wm e u + e λ π u1 un etu 1 un etw 1 wm e u + e λ π Huizhen u (U.H.) Bayesian : Message Passing in Singly Connected Feb / 17 π u, () p(x x pa() ) l (x ) λ w, (x ) Huizhen u (U.H.) Bayesian : Message Passing in Singly Connected Feb / 17 Huizhen u (U.H.) Bayesian : in Singly Connected Feb / 25 Outline P(X Tw1,..., X Twm X ) = P(X Tw1 X ) P(X Twm X ). (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, Bayesian : Message Passing in Singly Connected Bayesian : Bayesian Message : PassingMessage SinglyPassing Connected in Singly Connected p `xpa() Outline, x Bayesian & e = : Message p( & epassing Tu ) p`x in Singly & e Connected x pa() p(e Tw x ). in Chains in Trees in Singly Connected lief P Loopy u pa() Huizhen u lief P Huizhen u Huizhen u lief P Huizhen u (U.H.) Bayesian : Bayesian : Message : Huizhen liefpassing Propagation umessage in Singly Passing Connected Connected in Singly Connected Dept. Computer Science, Uni. of Helsinki Feb / 25 Loopy B Dept. Computer Science, Uni. Dept. ofcomputer Helsinki Science, Uni. of Helsinki nce Conditi in Chains Probabilistic Dept. Computer Models, Science, Spring, Uni. of Helsinki 2010 lief Lo in Trees Bayesian : Message Passing Probabilistic in Singly Models, Connected Probabilistic Spring, 2010 Models, Spring, 2010 in Singly Connected Huizhen u In SinglyHuizhen Connected u lief Co Huizhen u (U.H.) Bayesian : Message Passing in Singly Connected Feb / 17 nce Engine: O Loopy lief lief Propagat Message Passing Dept. Computer in a Science, Singly Connected Network Dept. Uni. Computer of Helsinki Loopy Science, Uni. of Helsinki in Chains Huizhen u lief Propagat Condi What do parents Outline need from in order to calculate their marginals? lief Propagat Huizhen u (U.H.) Bayesian Probabilistic : Message Models, Passing Probabilistic Spring, Singly Connected Models, 2010 Feb / 17 Huizhen u (U. Spring, 2010 Huizhen u (U.H.) Huizhen Bayesian u (U.H.) : Message Passing Bayesian in Singly : Connected Message Passing in Singly Connected Feb / 17 Feb / 17 A parent u needsdept. the Computer likelihoods Science, Uni. of Helsinki all based on the partial eidence e Loopy lief Pr Tu from the sub-polytree on s side with respect to u: Huizhen u Huizhen u (U.H.) Bayesian : Message λ Passing in Singly Connected Feb / 17,u() = X (U.H.) nce Bayesian Engine: : Oeriew Message Passing in Singly Connected Feb / 17 Huizhen Probabilistic X Models, Spring, p`x & e x pa() p( & e ) nce Engine: 2010Oeriew Tu p(e Tw x ) x in Chains x pa()\{u} u pa()\{u} nce Engine: in Oeriew Trees in Chains = X Huizhen u (U.H.) Huizhen u Bayesian (U.H.) : Message Bayesian Passing: Singly Connected Message Passing in Singly Connected Feb / 17 p`x x in Singly Outline Connected pa() l (x ) π u Feb / 17 Huizhen u (, ( ) λ w, (x ). x x pa()\{u} u pa()\{u} Loopy in Chains in Trees Huizhen u (U.H.) Bayesian : Message Passing in Singly Connected Feb / 17 Form of eidence Form of eidence Form of eidence ncee Engine: Tu1 u n Oeriew w 1 w m e u + e λ π u 1 u n w 1 u 1 w m u n e u + Form of eidence ew 1 wλ m π e u + e λ π in Singly Connected Huizhen u (U.H.) Bayesian : Message Passing in Feb. u / 17 w Singly Connected 16 4 Loopy Form of eidence Form of eidence Form of eidence u w 1 wu m e u + w 1 e w m λ eπ u + e λ π e Tu1 u n e Tw1 w m e u + e λ π Huizhen u (U.H.) nce E nce E lief P lief P Loopy B nce Conditi lief Lo lief nce Engine: Co O lief lief Propagat Loopy lief Propagat Condi lief Propagat Huizhen u (U.H.) Bayesian : Message Passing in Singly Connected Feb / 17 Huizhen u (U.H.) Bayesian : Message Passing in Singly Connected Feb / 17 Huizhen u (U.H.) Huizhen Bayesian u (U.H.) : Message Passing Bayesian in Singly : Connected Message Passing Connected Feb / 17 in Singly Feb / 17 lief P Loopy lief Pr Huizhen u (U.H.) Bayesian : in Singly Connected Feb / 25

6 Message Passing in a Singly Connected Network What do children need from in order to calculate their marginals? A child w needs for all x, the probability of x and the partial eidence e Tw from the sub-polytree on s side with respect to w: p(x & e Tw ) = X p`x & e x pa() p` & e Tu p(e Tw x ) x pa() u pa() = p`x x pa() l (x ) π u, () λ w, (x ). x pa() u pa() Algorithm Summary Each node sends to each u of its parents λ,u() = X X p(x x pa() ) l (x ) x x pa()\{u} π u, ( ) λ w, (x ), ; u pa()\{u} sends to each w of its children π,w (x ) = λ w, (x ) X p(x x pa() ) l (x ) x pa() π u, (), x ; u pa() e Tu1 u 1 u n when receiing all messages from parents and children, calculates p(x & e) = λ w, (x ) X π u, () p(x x pa() ) l (x ), x, x pa() u pa() P(e) = X x p(x & e), p(x e) = p(x & e)/p(e). w 1 w m Message passing schemes: (i) Each node can send a message to a linked node if it has receied messages from all the other linked nodes. e Tw1 (ii) Each node can send updated messages to linked nodes wheneer it gets a new message from some node. Huizhen u (U.H.) Bayesian : in Singly Connected Feb / 25 Huizhen u (U.H.) Bayesian : in Singly Connected Feb / 25 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. Example of Noisy-Or Gate X X X 1 2 n 1 2 n p(y x 1,..., x n) ( Q i:xi =1 qi, if y = 0; = 1 Q i:x i =1 qi, if y = 1. = OR( 1, 2,... n ) Express the message π Xi, i (x i) in the ector 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. iq i+(1 p i) x i {0,1} Example of Noisy-Or Gate Suppose e : { = 1} is receied. 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 alues?) 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 : in Singly Connected Feb / 25 Huizhen u (U.H.) Bayesian : in Singly Connected Feb / 25

7 Generalizations and Further Reading Find most probable configurations: max x p(x & e) : G is not a tree. We condition on certain ariables to create seeral 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 ariants and conergence analysis are actie research topics. Further reading: 1. Judea Pearl. Probabilistic Reasoning in Intelligent Systems, Morgan Kaufmann, Chap. 4. Huizhen u (U.H.) Bayesian : in Singly Connected Feb / 25