Constrained Bayesian Networks: Theory, Optimization, and Applications
This work addresses the challenge of modeling and inference in domains with limited data or specific constraints, such as arms control, though it appears incremental as it builds on existing Bayesian network and optimization methods.
The paper tackles the problem of probabilistic inference in causal networks when data is scarce or constraints are needed, by introducing Constrained Bayesian Networks that allow symbolic probabilities and real-valued constraints, and demonstrates its application in an arms control case study with an implementation using the Z3 SMT solver.
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.