Approximating probabilistic inference in Bayesian belief networks is NP-hard

It is known that exact computation of conditional probabilities in belief networks is NP-hard. Many investigators in the AI community have tacitly assumed that algorithms for performing approximate inference with belief networks are of polynomial complexity. Indeed, special cases of approximate inference can be performed in time polynomial in the input size. However, we have discovered that the general problem of approximating conditional probabilities with belief networks, like exact inference, resides in the NP-hard complexity class. We develop a complexity analysis to elucidate the difficulty of approximate probabilistic inference.