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imprecise probabilities

Imprecise probability is a statistical framework and generalized theory of uncertainty that represents beliefs or likelihoods using sets of probabilities, intervals, or bounds rather than a single exact numerical value. Unlike standard probability theory, which requires assigning a single precise probability distribution to an event, imprecise probability models can represent partial ignorance, incomplete evidence, or conflicting data by defining lower and upper probability limits or credal sets of plausible distributions. This approach encompasses various mathematical formalisms, including belief functions, possibility theory, and coherent lower previsions, making it especially useful in decision theory and machine learning for distinguishing epistemic uncertainty caused by a lack of knowledge from aleatoric uncertainty inherent in stochastic processes.

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Aleatoric and epistemic uncertainty in machine learning: an introduction to concepts and methods

Aleatoric and epistemic uncertainty in machine learning: an introduction to concepts and methods

Eyke Hüllermeier, Willem Waegeman

OrganizationsGhent UniversityPaderborn University

Why you should read this

Clarifies the critical distinction between irreducible data randomness and reducible model ignorance, providing a comprehensive framework for quantifying both aleatoric and epistemic uncertainty to build safer, more reliable machine learning systems.

The notion of uncertainty is of major importance in machine learning and constitutes a key element of machine learning methodology. In line with the statistical tradition, uncertainty has long been perceived as almost synonymous with standard probability and probabilistic predictions. Yet, due to the steadily increasing relevance of machine learning for practical applications and related issues such as safety requirements, new problems and challenges have recently been identified by machine learning scholars, and these problems may call for new methodological developments. In particular, this includes the importance of distinguishing between (at least) two different types of uncertainty, often referred to as aleatoric and epistemic. In this paper, we provide an introduction to the topic of uncertainty in machine learning as well as an overview of attempts so far at handling uncertainty in general and formalizing this distinction in particular.

Added

2026-09-16