Without any doubt, the notion of uncertainty is of major importance in machine learning and constitutes a key element of modern machine learning methodology. In recent years, it has gained in importance due to the increasing relevance of machine learning for practical applications, many of which are coming with safety requirements. In this regard, new problems and challenges have been identified by machine learning scholars, many of which call for novel methodological developments. Indeed, while uncertainty has a long tradition in statistics, and a broad range of useful concepts for representing and quantifying uncertainty have been developed on the basis of probability theory, recent research has gone beyond traditional approaches and also leverages more general formalism and uncertainty calculi.

In the statistics literature, two inherently different sources of uncertainty are commonly distinguished, referred to as aleatoric and epistemic. While the former refers to variability due to inherently random effects, the latter is uncertainty caused by a lack of knowledge and hence relates to the epistemic state of an agent. Thus, epistemic uncertainty can in principle be reduced on the basis of additional information, while aleatoric uncertainty is non-reducible. The distinction between different types of uncertainty and their quantification has also been adopted in the recent ML literature, and various methods for quantifying aleatoric and epistemic uncertainty have been proposed. In the context of supervised learning, the focus is typically on predictive uncertainty, i.e., the learner’s uncertainty in the outcome given a query instance for which a prediction is sought. The aleatoric part of this uncertainty is due to the supposedly stochastic nature of the dependence between instances and outcomes, e.g. due to a lack of informative features or noisy class annotations. Therefore, the “ground-truth” is a conditional probability distribution, i.e., each outcome has a certain probability to occur. Even with full knowledge about the underlying data-generating process, the outcome cannot be predicted with certainty.

Obviously, the learner does not have full knowledge of this ground-truth. Instead, it produces a “guess” on the basis of the sample data provided for training. Broadly speaking, epistemic uncertainty is uncertainty about the true probability and hence the discrepancy between the ground-truth and the learner's guess. This (second-order) uncertainty can be captured and represented in different ways. One approach is to train a probabilistic predictor in a more or less standard way, and to quantify the uncertainty of that predictor in a kind of post hoc manner. An example is the estimation of epistemic uncertainty in terms of the mutual information between the target variable and the model parameters.

Another idea is to estimate uncertainty in a more direct way, and to let the learner itself predict, not only the target variable, but also its own uncertainty about the prediction. For example, instead of predicting a probability distribution, the learner may predict a second-order distribution (distribution of distributions) or a set of distributions. Its epistemic uncertainty is then represented by the “peakedness” of the former or the size of the latter.  

This tutorial aims to provide an overview of uncertainty quantification in machine learning, a topic that has received increasing attention in the recent past. Starting with a recapitulation of classical ML concepts to represent uncertainty, wee specifically focus on novel approaches for distinguishing and representing so-called aleatoric and epistemic uncertainty. By the end of the tutorial, attendees will have a comprehensive understanding of the fundamental concepts and recent advances in this field.

PRESENTERS

This tutorial will be held in-person at IJCAI 2026 in Bremern. The program schedule will be as follows.

1. Motivation for uncertainty disentanglement (15 minutes)

2. What are the sources of aleatoric and epistemic uncertainty? (15 minutes)

2. Applications of uncertainty disentanglement (classification with reject option, active learning, active feature acquisition, etc., 30 minutes)

4. Short recap of methods that are unable to disentangle aleatoric and epistemic uncertainty (probabilistic models, calibration, conformal prediction, etc., 15 minutes)

5. Bayesian inference and ensemble methods (30 minutes)

6. Evidential deep learning methods (30 minutes)

7. Distance-based and density-based methods (15 minutes)

8. Credal sets and imprecise probabilities (15 minutes)

9. Critical notes on uncertainty disentanglement (30 minutes)

10. Open problems in uncertainty disentanglement (15 minutes)

Slides will be made available the evening before the event. 

For registration, see the IJCAI websige.