A Logical Approach for Factoring Belief Networks

We have shown recently that a belief network can be represented as a polynomial and that many probabilistic queries can be recovered in constant time from the partial derivatives of such a polynomial. Although this polynomial is exponential in size, we have shown that it can be “computed” using an arithmetic circuit whose size is not necessarily exponential. Hence, the key computational question becomes that of generating the smallest arithmetic circuit that computes the network polynomial, since an arithmetic circuit can be evaluated and all its partial derivatives computed in linear time. In this paper, we show that the process of generating an arithmetic circuit can be reduced to a process of transforming a propositional theory from one form into another. Specifically, we show that the network polynomial can be encoded efficiently using a propositional theory in Conjunctive Normal Form (CNF). We then show that if the CNF encoding is compiled into a Negation Normal Form (NNF) that satisfies three properties (smoothness, determinism, and decomposability), then one can extract from it in linear space an arithmetic circuit that computes the encoded polynomial. We discuss the merits of the proposed approach and present experimental results showing how it allows us to perform inference on belief networks that are intractable to structure-based methods for probabilistic inference.