Paul Dagum

Approximating probabilistic inference in Bayesian belief networks is NP-hard

Abstract 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. More specifically, we show that the existence of a polynomial-time relative approximation algorithm for major classes of problem instances implies that NP ⊆ P. We present our proof and explore the implications of the result.

An Optimal Algorithm for Monte Carlo Estimation

A typical approach to estimate an unknown quantity

Dynamic network models for forecasting

\mu

Restricted posterior leaflet motion after mitral ring annuloplasty

is to design an experiment that produces a random variable Z, distributed in [0,1] with E[Z]=\mu

An optimal approximation algorithm for Bayesian inference

, run this experiment independently a number of times, and use the average of the outcomes as the estimate. In this paper, we consider the case when no a priori information about Z is known except that is distributed in [0,1]. We describe an approximation algorithm

Mitral annular size and shape in sheep with annuloplasty rings

{\cal A}{\cal A}

Approximating the permanent of graphs with large factors

which, given

Mitral Leaflet Remodeling in Dilated Cardiomyopathy

\epsilon

Influence of Anterior Mitral Leaflet Second-Order Chordae Tendineae on Left Ventricular Systolic Function

and

Polytopes, permanents and graphs with large factors

\delta