The Computational Complexity of Probabilistic Inference Technical Report

The Inference problem in probabilistic or Bayesian networks—given a probabilistic network and a joint value assignment to a subset of its variables, compute the posterior probability of that assignment—is arguably the canonical computational problem related to such networks. Since it has been proven NP-hard by Cooper in 1990 [2], a number of complexity results have been obtained for the inference problem. For example, the question whether the posterior probability is non-zero is NP-complete [2] and whether it is larger than a threshold q is PP-complete [10]. Roth [14] established #P-completeness for the functional variant (i.e., where the actual probability is computed). These results are based on different reductions and often lack details (e.g., full membership proofs) due to space constraints. In this paper we discuss a unifying construction that reduces each of these problems from the corresponding Satisfiability variant and give membership proofs in full detail, allowing the reader to get a good understanding of these results and some important aspects of Probabilistic Turing Machines, which are the building blocks of these results. In addition, we show that the threshold variant of probabilistic inference, while fixed-parameter tractable for bounded treewidth, remains para-PP-complete even when the threshold is arbitrarily close to 1.