MAP Complexity Results and Approximation Methods

MAP is the problem of nding a most probable instantiation of a set of variables in a Bayesian network, given some evidence. MAP appears to be a signiicantly harder problem than the related problems of computing the probability of evidence (Pr), or MPE (a special case of MAP). Because of the complexity of MAP, and the lack of viable algorithms to approximate it, MAP computations are generally avoided by practitioners. This paper investigates the complexity of MAP. We show that MAP is complete for NP PP. We also provide negative complexity results for elimination based algorithms. It turns out that MAP remains hard even when MPE, and Pr are easy. We show that MAP is NP-complete when the networks are restricted to polytrees, and even then can not be eeectively approximated. Because there is no approximation algorithm with guaranteed results, we investigate best eeort approximations. We introduce a generic MAP approximation framework. As one instantiation of it, we implement local search coupled with belief propagation (BP) to approximate MAP. We show how to extract approximate evidence retraction information from belief propagation which allows us to perform eecient local search. This allows MAP approximation even on networks that are too complex to even exactly solve the easier problems of computing Pr or MPE. Experimental results indicate that using BP and local search provides accurate MAP estimates in many cases.