Pearl s algorithm

1. Pearl�s algorithm Message passing algorithm for exact inference in polytree BBNs Tomas Singliar, tomas@cs.pitt.edu

2. Outline Bayesian belief networks The inference task Idea of belief propagation Messages and incorporation of evidence Combining messages into belief Computing messages Algorithm outlined What if the BBN is not a polytree?

3. Bayesian belief networks (G, P) directed acyclic graph with the joint p.d. P each node is a variable of a multivariatedistribution links represent causal dependencies CPT in each node d-separation corresponds to independence polytree at most one path between Vi and Vk implies each node separates the graphinto two disjoint components (this graph is not a polytree) This slide introduces BBNs and the notion of polytree.This slide introduces BBNs and the notion of polytree.

4. The inference task we observe the values of some variables they are the evidence variables E Inference - to compute the conditional probability P(Xi |E)for all non-evidential nodes Xi Exact inference algorithm Variable elimination Join tree Belief propagation Computationally intensive � (NP-hard) Approximate algorithms Can have less than all nodes to compute (goal oriented).Can have less than all nodes to compute (goal oriented).

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

6. Pearl�s Belief Propagation Arrows are to indicate that this is a general situation.Arrows are to indicate that this is a general situation.

7. The ? Messages What are the messages? For simplicity, let the nodes be binary Let�s figure out what the messages are. Make the reservation that it is not exactly CPT but evidence needs to be incorporated. What is a CPT? It captures what we should believe the value of the node is. Based on parent information � may involve (causal) evidence.Let�s figure out what the messages are. Make the reservation that it is not exactly CPT but evidence needs to be incorporated. What is a CPT? It captures what we should believe the value of the node is. Based on parent information � may involve (causal) evidence.

8. The Evidence Evidence � values of observed nodes V3 = T, V6 = 3 Our belief in what the value of Vi�should� be changes. This belief is propagated As if the CPTs became

9. The Messages We know what the ? messages are What about ?? The messages are ?(V)=P(V|E+) and ?(V)=P(E-|V)

10. Combination of evidence Recall that EV = EV+ ? EV- and let us compute a is the normalization constant normalization is not necessary (can do it at the end) but may prevent numerical underflow problems

11. Messages Assume X received ?-messages from neighbors How to compute ?(x) = p(e-|x)? Let Y1, � , Yc be the children of X ?XY(x) denotes the ?-message sent between X and Y Discuss the formulas in depth � derive on board.Discuss the formulas in depth � derive on board.

12. Messages Assume X received p -messages from neighbors How to compute p(x) = p(x|e+) ? Let U1, � , Up be the parents of X pXY(x) denotes the p-message sent between X and Y summation over the CPT derive on boardderive on board

13. Messages to pass We need to compute pXY(x) Similarly, ?XY(x), X is parent, Y child Symbolically, group other parents of Y into V = V1, � , Vq

14. The Pearl Belief Propagation Algorithm We can summarize the algorithm now: Initialization step For all Vi=ei in E: ?(xi) = 1 wherever xi = ei ; 0 otherwise ?(xi) = 1 wherever xi = ei ; 0 otherwise For nodes without parents ?(xi) = p(xi) - prior probabilities For nodes without children ?(xi) = 1 uniformly (normalize at end)

15. The Pearl Belief Propagation Algorithm Iterate until no change occurs (For each node X) if X has received all the p messages from its parents, calculate p(x) (For each node X) if X has received all the ? messages from its children, calculate ?(x) (For each node X) if p(x) has been calculated and X received all the ?-messages from all its children (except Y), calculate pXY(x) and send it to Y. (For each node X) if ?(x) has been calculated and X received all the p-messages from all parents (except U), calculate ?XU(x) and send it to U. Compute BEL(X) = ?(x)p(x) and normalize in step 3, of course the message from Y may have been received too. analogously in step 4in step 3, of course the message from Y may have been received too. analogously in step 4

16. 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

17. 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

18. Thank you Questions welcome References Pearl, J. : Probabilistic reasoning in intelligent systems � Networks of plausible inference, Morgan � Kaufmann 1988 Castillo, E., Gutierrez, J. M., Hadi, A. S. : Expert Systems and Probabilistic Network Models, Springer 1997 Derivations shown in class are from this book, except that we worked with p instead of ? messages. They are related by factor of p(e+). www.cs.kun.nl/~peterl/teaching/CS45CI/bbn3-4.ps.gz Murphy, K.P., Weiss, Y., Jordan, M. : Loopy belief propagation for approximate inference � an empirical study, UAI 99