A recurrence local computation approach towards ordering composite beliefs in bayesian belief networks

Finding the l Most Probable Explanations (MPE) of a given evidence, S e , in a Bayesian belief network can be formulated as identifying and ordering a set of composite hypotheses, H i s, of which the posterior probabilities are the l largest; i.e., Pr(H 1 jS e) ::: Pr(H l jS e). When an order includes all the composite hypotheses in the network in order to nd all the probable explanations, it becomes a total order and the derivation of such an order has an exponential complexity. The focus of this paper is on the derivation of a partial order, with length l, for nding the l most probable composite hypotheses; where l typically is much smaller than the total number of composite hypotheses in a network. Previously, only the partial order of length two (i.e., l = 2) in a singly connected Bayesian network could be eeciently derived without further restriction on network topologies and the increase in spatial complexity. This paper discusses an eecient algorithm for the derivation of the partial ordering of the composite hypotheses in a singly connected network with arbitrary order length. This algorithm is based on the propagation of quantitative vector streams in a feed-forward manner to a designated \root" node in a network. The time complexity of the algorithm is in the order of O(lkn); where l is the length of a partial order, k the length of the longest path in a network, and n the maximum number of node states | deened as the product of the size of the conditional probability table of a node and the number of incoming messages towards the node.