Generalized Weighted Model Counting: An Efficient Monte-Carlo Meta-Algorithm

In this paper, we focus on computing the prices of secu- rities represented by logical formulas in combinatorial prediction markets when the price function is represented by a Bayesian network. This problem turns out to be a natural extension of the weighted model counting (WMC) problem (Sang, Bearne, and Kautz 2005), which we call generalized weighted model counting (GWMC) problem. In GWMC, we are given a logical formula F and a polynomial-time computable weight function. We are asked to compute the total weight of the valuations that satisfy F. Based on importance sampling, we propose a Monte-Carlo meta-algorithm that has a good theoretical guarantee for formulas in disjunctive normal form (DNF). The meta-algorithm queries an oracle algorithm that computes marginal probabilities in Bayesian networks, and has the following theoretical guarantee. When the weight function can be approximately represented by a Bayesian network for which the oracle algorithm runs in polynomial time, our meta-algorithm becomes a fully polynomial-time randomized approximation scheme (FPRAS).