A New Characterization of Probabilities in Bayesian Networks

TL;DR

This paper introduces quasi-probabilities as an algebraic framework for Bayesian networks, enabling efficient computation and manipulation of probabilities through algebraic and signal processing methods.

Contribution

It presents a novel algebraic characterization of Bayesian network probabilities using quasi-probabilities derived from noisy AND-OR-NOT networks.

Findings

Quasi-probabilities allow recursive computation of marginal probabilities.

Joint probabilities can be obtained by multiplying quasi-probabilities.

Pulse train representations facilitate efficient probabilistic inference.

Abstract

We characterize probabilities in Bayesian networks in terms of algebraic expressions called quasi-probabilities. These are arrived at by casting Bayesian networks as noisy AND-OR-NOT networks, and viewing the subnetworks that lead to a node as arguments for or against a node. Quasi-probabilities are in a sense the "natural" algebra of Bayesian networks: we can easily compute the marginal quasi-probability of any node recursively, in a compact form; and we can obtain the joint quasi-probability of any set of nodes by multiplying their marginals (using an idempotent product operator). Quasi-probabilities are easily manipulated to improve the efficiency of probabilistic inference. They also turn out to be representable as square-wave pulse trains, and joint and marginal distributions can be computed by multiplication and complementation of pulse trains.