math.ST · 2605.07908 Copy arXiv ID · May 8, 2026 Save Statistical inference with belief functions: A survey Authors: Fabio Cuzzolin
Organizations: Institute for AI, Data Analysis and Systems (AIDAS) · School of Engineering, Computing & Mathematics, Oxford Brookes University
Abstract Belief functions are a powerful and popular framework for the mathematical characterisation of uncertainty, in particular in situations in which lack of data renders learning a probability distribution for the problem impractical. The first step in a reasoning chain based on belief functions is inference: how to learn a belief measure from the available data. In this survey we focus, in particular, on making inference from statistical data, and review the most significant contributions in the area.
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Classical deep learning typically operates on individual cases. Despite its success, real-world usage often requires repeated inference to estimate statistical quantities for complex decision-making tasks involving uncertainty or extreme-value analysis, resulting in substantial latency. We introduce neural statistical functions, a new family of models learned from pre-trained single-sample predictors and scattered data samples, which can directly infer statistics over continuous operating condition ranges without explicit sampling. By introducing the notion of prefix statistics, we transform and unify diverse statistical functions (e.g., integrals, quantiles, and maxima) into an interval-conditional framework, in which a principled identity between the prefix statistics and the individual-case regression serves as the learning objective. Neural statistical functions achieve strong performance in estimating essential statistics of complex physical processes, including accumulated energy in dynamical systems, quantiles of aerodynamic responses, and maximum stress in crash processes, while achieving up to a 100
× \times × reduction in model evaluations.