Finding MAPs for Belief Networks is NP-Hard

Given a probabilistic world model, an important problem is to find the maximum a-posteriori probability (MAP) instantiation of all the random variables given the evidence. Numerous researchers using such models employ some graph representation for the distributions, such as a Bayesian belief network. This representation simplifies the complexity of specifying the distributions from exponential in n, the number of variables in the model, to linear in n, in many interesting cases. We show, however, that finding the MAP is NP-hard in the general case when these representations are used, even if the size of the representation happens to be linear in n. Furthermore, minor modifications to the proof show that the problem remains NP-hard for various restrictions of the topology of the graphs. The same technique can be applied to the results of a related paper (by Cooper), to further restrict belief network topology in the proof that probabilistic inference is NP-hard.