Automatically Finding the Right Probabilities in Bayesian Networks

Abstract

This paper presents alternative techniques for inference on classical Bayesian networks in which all probabilities are fixed, and for synthesis problems when conditional probability tables (CPTs) in such networks contain symbolic parameters rather than concrete probabilities. The key idea is to exploit probabilistic model checking as well as its recent extension to parameter synthesis techniques for parametric Markov chains. To enable this, the Bayesian networks are transformed into Markov chains and their objectives are mapped onto probabilistic temporal logic formulas. For exact inference, we compare probabilistic model checking to weighted model counting on various Bayesian network benchmarks. We contrast symbolic model checking using multi-terminal binary (aka: algebraic) decision diagrams to symbolic inference using proba- bilistic sentential decision diagrams, symbolic data structures that are tailored to Bayesian networks. For the parametric setting, we describe how our techniques can be used for various synthesis problems such as computing sensitivity functions (and values), simple and difference parameter tuning and ratio parameter tuning. Our parameter synthesis techniques are applicable to arbitrarily many, possibly dependent, parameters that may occur in multiple CPTs. This lifts restrictions, e.g., on the number of parametrized CPTs, or on parameter dependencies between several CPTs, that exist in the literature. Experiments on several benchmarks show that our parameter synthesis techniques can treat parameter synthesis for Bayesian networks (with hundreds of unknown parameters) that are out of reach for existing techniques.

Parameter Synthesis in Markov Models: A Gentle Survey

“…Beyond Markov models, probabilistic graphical models in general and Bayesian networks in particular are widespread to describe complex conditional probability distributions. Recent work [72,73] shows that ideas and methods for parameter synthesis in Markov chains as described in this survey significantly improve upon existing methods for parametric Bayesian networks [21]. Vice versa, some inference techniques do yield interesting alternatives for the analysis of (finite-horizon properties in) pMCs [46].…”

“…There can be substantial differences between the heuristics for candidate generation, especially in settings where single region verification calls are expensive. To indicate the scalability of the approach, we report on some recent experiments [73] on parametric Bayesian networks. Figure 12 shows that 80-90% of the 8-dimensional parameter space for three parametric Bayesian networks (win95pts, hailfinder, and hepar-2) can be covered in about 100-1,000 s.…”