An Optimal Approximation Algorithm for Bayesian Inference

Approximating the inference probability Pr X xjE e in any sense even for a single evidence node E is NP hard This result holds for belief networks that are allowed to contain extreme conditional probabilities that is conditional probabilities arbitrarily close to Nevertheless all previous approximation algorithms have failed to approximate e ciently many inferences even for belief networks without extreme conditional probabilities We prove that we can approximate e ciently probabilistic inference in belief net works without extreme conditional probabilities We construct a randomized approx imation algorithm the bounded variance algorithm that is a variant of the known likelihood weighting algorithm The bounded variance algorithm is the rst algorithm with provably fast inference approximation on all belief networks without extreme con ditional probabilities From the bounded variance algorithm we construct a deterministic approximation algorithm using current advances in the theory of pseudorandom generators In contrast to the exponential worst case behavior of all previous deterministic approximations the deterministic bounded variance algorithm approximates inference probabilities in worst case time that is subexponential logn d for some integer d that is a linear function of the depth of the belief network