Constrained Bayesian Networks: Theory, Optimization, and Applications
  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.
