Robustness Guarantees for Credal Bayesian Networks via Constraint Relaxation over Probabilistic Circuits

TL;DR

This paper introduces a scalable method to compute robustness guarantees for credal Bayesian networks by relaxing constraints in probabilistic circuits, providing guaranteed bounds on worst-case probabilities under uncertainty.

Contribution

It develops a linear-time relaxation technique for the MARmax problem in credal Bayesian networks, enabling efficient robustness analysis with theoretical insights into bound tightness.

Findings

The method provides near-tight upper bounds in experiments.

It scales better than existing approaches.

The relaxation's tightness relates to network structure.

Abstract

In many domains, worst-case guarantees on the performance (e.g., prediction accuracy) of a decision function subject to distributional shifts and uncertainty about the environment are crucial. In this work we develop a method to quantify the robustness of decision functions with respect to credal Bayesian networks, formal parametric models of the environment where uncertainty is expressed through credal sets on the parameters. In particular, we address the maximum marginal probability (MARmax) problem, that is, determining the greatest probability of an event (such as misclassification) obtainable for parameters in the credal set. We develop a method to faithfully transfer the problem into a constrained optimization problem on a probabilistic circuit. By performing a simple constraint relaxation, we show how to obtain a guaranteed upper bound on MARmax in linear time in the size of the circuit. We further theoretically characterize this constraint relaxation in terms of the original Bayesian network structure, which yields insight into the tightness of the bound. We implement the method and provide experimental evidence that the upper bound is often near tight and demonstrates improved scalability compared to other methods.