  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.
