Global Sensitivity Analysis of Uncertain Parameters in Bayesian Networks

TL;DR

This paper addresses the insufficiency of one-at-a-time (OAT) sensitivity analyses for Bayesian networks by introducing a global, variance-based framework based on Sobol indices. It encodes uncertain CPT entries as additional BN variables and uses tensor-train factorization to manage the resulting high dimensionality, enabling exact Sobol-based sensitivity analysis via TN contraction. The approach reveals substantial higher-order interactions that OAT misses, demonstrated on a COVID-19 measures BN where Sobol indices differ markedly from local sensitivities and interactions can dominate individual effects. The method offers a scalable, interpretable tool for uncertainty quantification in complex BN models with practical implications for robustness assessment and decision support.

Abstract

Traditionally, the sensitivity analysis of a Bayesian network studies the impact of individually modifying the entries of its conditional probability tables in a one-at-a-time (OAT) fashion. However, this approach fails to give a comprehensive account of each inputs' relevance, since simultaneous perturbations in two or more parameters often entail higher-order effects that cannot be captured by an OAT analysis. We propose to conduct global variance-based sensitivity analysis instead, whereby $n$ parameters are viewed as uncertain at once and their importance is assessed jointly. Our method works by encoding the uncertainties as $n$ additional variables of the network. To prevent the curse of dimensionality while adding these dimensions, we use low-rank tensor decomposition to break down the new potentials into smaller factors. Last, we apply the method of Sobol to the resulting network to obtain $n$ global sensitivity indices. Using a benchmark array of both expert-elicited and learned Bayesian networks, we demonstrate that the Sobol indices can significantly differ from the OAT indices, thus revealing the true influence of uncertain parameters and their interactions.

Figures (10)

• Figure 1: An example of a DAG over three binary random variables $Y_1,Y_2,Y_3$ with the associated probability specifications $\boldsymbol{\theta}^0$.
• Figure 2: Sensitivity functions for the output probability $P(Y_3 = \textnormal{yes})$ as a function of $P(Y_1=\textnormal{yes})$ (left) and $P(Y_2=\textnormal{yes})$ (right).
• Figure 3: Two-way sensitivity function for $P(Y_3 = \textnormal{yes})$ as a function of $P(Y_1=\textnormal{yes})$ and $P(Y_2=\textnormal{yes})$.
• Figure 4: The asia network as a Bayesian network (a), Markov random field (b), and tensor network (c). Note that, if the cliques of (b) are regarded as hyperedges, then (c) is the dual graph of (b).
• Figure 5: Encoding four uncertainties into a 3D CPT results in a new potential of dimensionality $3 + 4 = 7$, which we compactly express using the TT format. The new indices $R_k$ and $S_k$ are virtual (in a probabilistic interpretation, they can be thought of as latent variables) and only serve to factorize the original tensor into smaller-dimensional tensors, thus avoiding an exponential increase in the number of parameters.

Theorems & Definitions (2)

• Lemma 1
• proof