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A belief network comprises a graphical representation of dependencies between variables of a domain and a set of conditional probabilities associated with each dependency. Unless P=NP, an efficient, exact algorithm does not exist to compute probabilistic inference in belief networks. Stochastic simulation methods, which often improve run times, provide an alternative to exact inference algorithms. We present such a stochastic simulation algorithm 2)-BNRAS that is a randomized approximation scheme. To analyze the run time, we parameterize belief networks by the dependence value P E , which is a measure of the cumulative strengths of the belief network dependencies given background evidence E. This parameterization defines the class of f-dependence networks. The run time of 2)-BNRAS is polynomial when f is a polynomial function. Thus, the results of this paper prove the existence of a class of belief networks for which inference approximation is polynomial and, hence, provably faster than any exact algorithm.

Researchers in decision analysis and artificial intelligence (AI) have used Bayesian belief networks to build probabilistic expert systems. Using standard methods drawn from the theory of computational complexity, workers in the field have shown that the problem of probabilistic inference in belief networks is difficult and almost certainly intractable. We have developed a randomized approximation scheme, BN-RAS, for doing probabilistic inference in belief networks. The algorithm can, in many circumstances, perform efficient approximate inference in large and richly interconnected models. Unlike previously described stochastic algorithms for probabilistic inference, the randomized approximation scheme (ras) computes a priori bounds on running time by analyzing the structure and contents of the belief network. In this article, we describe BN-RAS precisely and analyze its performance mathematically.

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In recent years, researchers in decision analysis and arti ficial intelligence (AI) have used Bayesian belief networks to build models of expert opinion. Using standard meth ods drawn from the theory of computational complex ity, workers in the field have shown that the problem of probabilistic inference in belief networks is difficult and almost certainly intractable. KNET, a software environ ment for constructing knowledge-based systems within the axiomatic framework of decision theory, contains a randomized approximation scheme for probabilistic infer ence. The algorithm can, in many circumstances, per form efficient approximate inference in large and richly interconnected models of medical diagnosis. Unlike previ ously described stochastic algorithms for probabilistic in ference, the randomized approximation scheme computes a priori bounds on running time by analyzing the struc ture and contents of the belief network.In this article, we describe a randomized algorithm for probabilistic inference and analyze its performance math ematically. Then, we devote the major portion of the paper to a discussion of the algorithm's empirical behav ior. The results indicate that the generation of good trials (that is, trials whose distribution closely matches the true distribution) , rather than the computation of numerous mediocre trials, dominates the performance of stochastic simulation.

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