Large Deviation Methods for Approximate Probabilistic Inference
This provides a theoretical framework for approximate inference in large networks, addressing a bottleneck in probabilistic modeling for researchers in machine learning and statistics.
The paper tackles the problem of intractable exact probabilistic inference in large two-layer belief networks with binary variables and monotonic conditional probabilities, deriving rigorous upper and lower bounds on marginal probabilities using large deviation theory and proving convergence rates as network size increases.
We study two-layer belief networks of binary random variables in which the conditional probabilities Pr[childlparents] depend monotonically on weighted sums of the parents. In large networks where exact probabilistic inference is intractable, we show how to compute upper and lower bounds on many probabilities of interest. In particular, using methods from large deviation theory, we derive rigorous bounds on marginal probabilities such as Pr[children] and prove rates of convergence for the accuracy of our bounds as a function of network size. Our results apply to networks with generic transfer function parameterizations of the conditional probability tables, such as sigmoid and noisy-OR. They also explicitly illustrate the types of averaging behavior that can simplify the problem of inference in large networks.