Reasoning MPE to Multiply Connected Belief Networks Using Message Passing

Finding the I Most Probable IJxplanations (MPE) of a given evidence, Se, in a Bayesian belief network is a process to identify and order a set of composite hypotheses, His, of which the posterior probabilities are the I largest; i.e., Pr(Hii&) 2 Pr(H21&) 2 . . . L Pr(&ISlz)~ A composite hypothesis is defined as an instantiation of all the non-evidence variables in the network. It could be shown that finding all the probable explanations is a NP-hard problem. Previously, only the first two best explanations (i.e., I = 2) in a singly connected Bayesian network could be efficiently derived without restrictions on network topologies and probability distributions. This paper presents an efficient algorithm for finding d (2 2) MPE in singlyconnected networks and the extension of this algorithm for multiply-connected networks. This algorithm is based on a message passing scheme and has a time complexity O(lkn) for singly-connected networks; where I is the number of MPE to be derived, k the length of the longest path in a network, and n the maximum number of node states defined as the product of the size of the conditional probability table of a node and the number of the incoming/outgoing arcs of the node. Whenever a variable in a Bayesian network is observable, this variable is referred to as an evidence variable. The set of evidence variables is represented by S,. Given a Se, an instantiation of all the non-evidence variables H in a Bayesian belief network is referred to as a composite hypothesis. Each H is said to be a probable explanation of the given observation, S,, in a Bayesian belief network if Pr(HjS,) > 0. Finding the I Most Probable &xplanations (MPE) of a given evidence, Se, in a Bayesian belief network is to identify and order a set of composite hypotheses, His, of which the posterior probabilities are the 2 largest; i.e., J+(&]S,) 2 Pr(HzlS,) L . . . L P+&]SJ.