Sensitivity analysis, multilinearity and beyond
This work provides a theoretical foundation for sensitivity methods in probabilistic models, extending applicability to context-specific, dynamic Bayesian networks, and chain event graphs, but it is incremental as it builds on existing sensitivity analysis frameworks.
The paper tackles the problem of sensitivity analysis in discrete Bayesian networks by developing a unifying algebraic approach based on the multilinear structure of atomic probabilities, proving that the Chan-Darwiche distance is minimized for certain multi-parameter variations when parameters are proportionally covaried.
Sensitivity methods for the analysis of the outputs of discrete Bayesian networks have been extensively studied and implemented in different software packages. These methods usually focus on the study of sensitivity functions and on the impact of a parameter change to the Chan-Darwiche distance. Although not fully recognized, the majority of these results heavily rely on the multilinear structure of atomic probabilities in terms of the conditional probability parameters associated with this type of network. By defining a statistical model through the polynomial expression of its associated defining conditional probabilities, we develop a unifying approach to sensitivity methods applicable to a large suite of models including extensions of Bayesian networks, for instance context-specific and dynamic ones, and chain event graphs. By then focusing on models whose defining polynomial is multilinear, our algebraic approach enables us to prove that the Chan-Darwiche distance is minimized for a certain class of multi-parameter contemporaneous variations when parameters are proportionally covaried.