Propagation in Poly Trees Given a Bayesian Network BN = {G, JDP} JDP(a,b,c,d,e) = p(a)*p(b|a)*p(c|e,b)*p(d)*p(e|d) a d b e c

Network Initialization The network is initialized by giving prior probabilities to root nodes and conditional probabilities (links) for all non- root nodes. a b c d e f g Values of a Values of d p(d|a)

Belief Update After initialization, the network is ready to receive evidence. (1) Direct evidence. Updates the node's  and vectors as well as the belief vector. Then it propagates  and messages to node's children and parents. (2) Causal evidence. Comes from parents, as  messages which act as node’s prior probability B A C D E F H G (3) Diagnostic evidence from children, as messages acts as the node’s likelihood vector,.

B A C D E F H G Example Direct evidence is given to D Probability vector in D is recomputed Messages sent to Parents And children Probability vector in A is recomputed Messages are sent Probability vector in B is recomputedProbability vector in C is recomputed Probability vector in E is recomputed Probability vector in F is recomputed Probability vector in G is recomputed Probability vector in H is recomputed

Computation of Belief Belief at each node is computed by: Bel(x) =  (x) *  (x) First the  and vectors are computed where  D (A) is the last  message sent to D from parent A. where k represents the k th message from the k th child After the and  vectors are updated, the node updates its belief, and is ready to propagate messages to its parents and children. The vector is computed as follows:

Computation of Messages The message that node D sends to parent A is computed as follows: The  message that node D sends to its children (e.g., child E): Or alternatively:

What about loops?? B A C D E F H G The algorithm fails with multiply connected networks