Propagation of~Interval Belief Structures and~Imprecise Copulas for~Neural Network Verification
2026-06-29 • Artificial Intelligence
Artificial IntelligenceLogic in Computer Science
AI summaryⓘ
The authors address the challenge of checking neural networks when the exact probabilities about inputs and their relationships are not fully known. They present a method that uses intervals and uncertain relationship models to handle this lack of precise information. Their approach propagates these uncertainties through the network, guaranteeing safe estimates for the output probabilities within known bounds. This helps ensure reliable verification even when input data is partially uncertain.
neural networksquantitative verificationimprecise probabilitiesinterval belief structurescopulasuncertainty propagationaffine transformationsactivation functionsprobabilistic safetypush-forward measure
Authors
Francesc Pifarre-Esquerda, Eric Goubault, Sylvie Putot
Abstract
Quantitative verification of neural networks requires reasoning about probabilities under substantial uncertainty in both input distributions and their dependence structure. In realistic settings, this information is often only partially specified, and assuming precise probabilistic models can lead to unreliable results. We propose a sound framework for quantitative verification under imprecise probabilistic information, combining interval belief structures to represent marginal uncertainty with imprecise copulas to model uncertain dependence. We develop a propagation method for imprecisely coupled interval belief structures through feed-forward neural networks. Using mixed imprecise copula volumes, we derive sound push-forward constructions through affine transformations and activation functions. The resulting output can provide guaranteed lower and upper bounds on probabilistic safety properties, valid for all probability models compatible with the specified imprecise inputs.