Constrained Bayesian Networks: Theory, Optimization, and Applications

TL;DR

This paper introduces Constrained Bayesian Networks, a flexible probabilistic modeling framework that incorporates application-specific constraints and symbolic probabilities, supported by formal semantics and optimization algorithms, demonstrated through an arms control case study.

Contribution

It develops the theory, semantics, and algorithms for Constrained Bayesian Networks, enabling inference with symbolic probabilities and constraints, especially useful with limited data.

Findings

Successfully modeled an arms control case study with limited data.

Implemented a prototype using Z3 SMT solver for decision procedures.

Demonstrated the approach's effectiveness through experimental evaluation.

Abstract

We develop the theory and practice of an approach to modelling and probabilistic inference in causal networks that is suitable when application-specific or analysis-specific constraints should inform such inference or when little or no data for the learning of causal network structure or probability values at nodes are available. Constrained Bayesian Networks generalize a Bayesian Network such that probabilities can be symbolic, arithmetic expressions and where the meaning of the network is constrained by finitely many formulas from the theory of the reals. A formal semantics for constrained Bayesian Networks over first-order logic of the reals is given, which enables non-linear and non-convex optimisation algorithms that rely on decision procedures for this logic, and supports the composition of several constrained Bayesian Networks. A non-trivial case study in arms control, where few or no data are available to assess the effectiveness of an arms inspection process, evaluates our approach. An open-access prototype implementation of these foundations and their algorithms uses the SMT solver Z3 as decision procedure, leverages an open-source package for Bayesian inference to symbolic computation, and is evaluated experimentally.