Solving MAP Exactly using Systematic Search

MAP is the problem of finding a most probable instantiation of a set of variables in a Bayesian network, given some partial evidence about the complement of that set. Unlike posterior probabilities, or MPE (a special case of MAP), the time and space complexity of structure–based algorithms for MAP are not only exponential in the network treewidth, but in a larger parameter known as the constrained treewidth. In practice, this means that computing MAP can be orders of magnitude more expensive than computing posterior probabilities or MPE. We introduce in this paper a new, simple upper bound on the probability of a MAP solution, which is shown to be generally much tighter than existing bounds. We then use the proposed upper bound to develop a branch–and– bound search algorithm for solving MAP exactly. Experimental results demonstrate that the search algorithm is able to solve many problems that are far beyond the reach of any structure–based method for MAP. For example, we show that the proposed algorithm can compute MAP exactly and efficiently for some networks whose constrained treewidth is more than 40.