Pearl’s Belief Propagation Algorithm Exact answers from tree-structured Bayesian networks Heavily based on slides by: Tomas Singliar, Modified by: Daniel Lowd, for UW CSE 573

Outline What’s a polytree? Idea of belief propagation Messages and incorporation of evidence Combining messages into belief Computing messages Algorithm outlined What if the BN is not a polytree?

Polytree Bayesian networks What is a polytree?  A Bayesian network graph is a polytree if (an only if) there is at most one path between any two nodes, V i and V k Why do we care about polytrees?  Exact BN inference is NP-hard…  …but on polytrees, takes linear time.

Examples: Polytree or not? V1V1 V5V5 V2V2 V4V4 V3V3 V6V6 V1V1 V5V5 V2V2 V4V4 V3V3 V6V6 V1V1 V2V2 V3V3 V4V4

Our inference task We know the values of some evidence variables E: We wish to compute the conditional probability P(X i |E) for all non-evidence variables X i.

Pearl’s belief propagation Local computation for one node V desired Information flows through the links of G  flows as messages of two types – λ and π V splits network into two disjoint parts  Strong independence assumptions induced – crucial! Denote E V + the part of evidence accessible through the parents of V (causal)  passed downward in π messages Analogously, let E V - be the diagnostic evidence  passed upwards in λ messages

Pearl’s Belief Propagation V U2U2 V1V1 V2V2 U1U1 π(U 2 ) π(V 1 ) π(V 2 ) π(U 1 ) λ(U 1 ) λ(V 2 ) λ(V 1 ) λ(U 2 )

The  Messages What are the messages? For simplicity, let the nodes be binary V1V1 V2V2 V 1 =T0.8 V 1 =F0.2 PV 1 =TV 1 =F V 2 =T V 2 =F The message passes on information. What information? Observe: P(V 2 | V 1 ) = P(V 2 | V 1 =T)P(V 1 =T) + P(V 2 | V 1 =F)P(V 1 =F) The information needed is the CPT of V 1 =  V (V 1 )  Messages capture information passed from parent to child

The Evidence Evidence – values of observed nodes  V 3 = T, V 6 = 3 Our belief in what the value of V i ‘should’ be changes. This belief is propagated As if the CPTs became V1V1 V5V5 V2V2 V4V4 V3V3 V6V6 V 3 =T1.0 V 3 =F0.0 PV 2 =TV 2 =F V 6 =10.0 V 6 =20.0 V 6 =31.0

The Messages We know what the  messages are What about ? The messages are  (V)=P(V|E + ) and (V)=P(E - |V) V1V1 V2V2 Assume E = { V 2 } and compute by Bayes rule: The information not available at V 1 is the P(V 2 |V 1 ). To be passed upwards by a -message. Again, this is not in general exactly the CPT, but the belief based on evidence down the tree.

Combination of evidence Recall that E V = E V +  E V - and let us compute α is the normalization constant normalization is not necessary (can do it at the end) but may prevent numerical underflow problems

Messages Assume X received λ-messages from neighbors How to compute λ(x) = p(e - |x)? Let Y 1, …, Y c be the children of X λ XY (x) denotes the λ-message sent between X and Y

Messages Assume X received π -messages from neighbors How to compute π(x) = p(x|e + ) ? Let U 1, …, U p be the parents of X π XY (x) denotes the π-message sent between X and Y summation over the CPT

Messages to pass We need to compute π XY (x) Similarly, λ XY (x), X is parent, Y child Symbolically, group other parents of Y into V = V 1, …, V q

The Pearl Belief Propagation Algorithm We can summarize the algorithm now:  Initialization step For all V i =e i in E:  (x i ) = 1 wherever x i = e i ; 0 otherwise   (x i ) = 1 wherever x i = e i ; 0 otherwise For nodes without parents   (x i ) = p(x i ) - prior probabilities For nodes without children  (x i ) = 1 uniformly (normalize at end)

The Pearl Belief Propagation Algorithm  Iterate until no change occurs (For each node X) if X has received all the π messages from its parents, calculate π(x) (For each node X) if X has received all the λ messages from its children, calculate λ(x) (For each node X) if π(x) has been calculated and X received all the λ-messages from all its children (except Y), calculate π XY (x) and send it to Y. (For each node X) if λ(x) has been calculated and X received all the π-messages from all parents (except U), calculate λ XU (x) and send it to U.  Compute BEL(X) = λ(x)π(x) and normalize

Complexity On a polytree, the BP algorithm converges in time proportional to diameter of network – at most linear Work done in a node is proportional to the size of CPT Hence BP is linear in number of network parameters For general BBNs  Exact inference is NP-hard  Approximate inference is NP-hard

Most Graphs are not Polytrees Cutset conditioning  Instantiate a node in cycle, absorb the value in child’s CPT.  Do it with all possible values and run belief propagation.  Sum over obtained conditionals  Hard to do Need to compute P(c) Exponential explosion - minimal cutset desirable (also NP-complete) Clustering algorithm Approximate inference  MCMC methods  Loopy BP