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

TL;DR

This work addresses robustness of discrete Bayesian networks to misspecification by moving beyond KL-based sensitivity and adopting total variation distance. It introduces the diameter as a succinct measure of dependence in a CPT and develops a suite of diameter-based sensitivity tools (edge strength, edge weighted influence, level amalgamation, asymmetry strength) to bound and prioritize perturbations without requiring a fully specified model. The framework yields provable error propagation bounds along junction-tree paths and provides practical elicitation guidance, illustrated through Asia and ISTAT networks and supported by implementation in bnmonitor. Overall, the approach offers transparent, scalable bounds for sensitivity analysis and robust BN elicitation, with potential extensions to dynamic and mixed variable settings.

Abstract

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. In this paper, we argue that robustness methods based instead on the familiar 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. We demonstrate how such formal robustness considerations can be embedded in building a Bayesian network.

Figures (9)

• Figure 1: Illustration of the triangulation of a BN.
• Figure 2: The istat BN learned over the 2012 ISTAT enterprise innovation data.
• Figure 3: The junction tree representing the asia BN. Initial of variable names are used. Cliques are circled, while separators are squared.
• Figure 4: Heatmap of the diameters of the istat BN. Darker colors represent higher diameter values.
• Figure 5: The asia BN with edge strengths as edges' labels and widths.

Theorems & Definitions (20)

• Definition 1
• Definition 2
• Proposition 1
• Proposition 2
• Proposition 3
• Definition 3
• Proposition 4
• Proposition 5
• Definition 4
• Definition 5