The diameter of a stochastic matrix: A new measure for sensitivity analysis in Bayesian networks

Bayesian networks are one of the most widely used classes of probabilistic models for risk management and decision support because of their interpretability and flexibility in including heterogeneous pieces of information. In any applied modelling, it is critical to assess how robust the inferences on certain target variables are to changes in the model. In Bayesian networks, these analyses fall under the umbrella of sensitivity analysis, which is most commonly carried out by quantifying dissimilarities using Kullback-Leibler information measures. We argue that robustness methods based instead on the total variation distance provide simple and more valuable bounds on robustness to misspecification, which are both formally justifiable and transparent. We introduce a novel measure of dependence in conditional probability tables called the diameter to derive such bounds. This measure quantifies the strength of dependence between a variable and its parents. Furthermore, the diameter is a versatile measure that can be applied to a wide range of sensitivity analysis tasks. It is particularly useful for quantifying edge strength, assessing influence between pairs of variables, detecting asymmetric dependence, and amalgamating levels of variables. This flexibility makes the diameter an invaluable tool for enhancing the robustness and interpretability of Bayesian network models in applied risk management and decision support.