Looking at the posterior - accuracy and uncertainty of neural-network predictions

Aug 3, 2026

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Bayesian inference enables neural networks to express predictive uncertainty through posterior distributions over model parameters and network outputs. By analyzing these posteriors, uncertainty can be decomposed into aleatoric and epistemic components, clarifying different sources of unreliability. The work demonstrates that prediction accuracy depends on both uncertainty types in a nuanced way that cannot be derived from marginalized distributions alone. The relationship varies with model architecture and dataset properties, and the paper proposes a new acquisition function for active learning that outperforms common uncertainty-based methods.

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Abstract

Introduction

Uncertainty quantification and surrogate accuracy measures

Bayesian posteriors and predictive distributions

Aleatoric vs epistemic uncertainty and predictive uncertainty

arXiv :2211 . 14605v2 [ cs .LG] 22 Nov 2023 Looking at the posterior: accuracy and uncertainty of neural-network predictions Hampus Linander 1 ,2‡, Oleksandr Balabanov3 , Henry Yang 1 , Bernhard Mehlig 1 1 Department of Physics, University of Gothenburg, 41296 Gothenburg, Sweden 2 Department of Mathematical Sciences, Chalmers University of Technology, University of Gothenburg, 41296 Gothenburg, Sweden 3 Department of Physics, Stockholm University, 10691 Stockholm, Sweden Abstract. Bayesian inference can quantify uncertainty in the predictions of neural networks using posterior distributions for model parameters and network output. By looking at these posterior distributions, one can separate the origin of uncertainty into aleatoric and epistemic contributions. One goal of uncertainty quantification is to inform on prediction accuracy. Here we show that prediction accuracy depends on both epistemic and aleatoric uncertainty in an intricate fashion that cannot be understood in terms of marginalized uncertainty distributions alone. How the accuracy relates to epistemic and aleatoric uncertainties depends not only on the model architecture, but also on the properties of the dataset. We discuss the significance of these results for active learning and introduce a novel acquisition function that outperforms common uncertainty-based methods. To arrive at our results, we approximated the posteriors using deep ensembles, for fully-connected, convolutional and attention-based neural networks. ‡ email: [linander@chalmers.se](linander@chalmers.se) Looking at the posterior: accuracy and uncertainty of neural-network predictions 2 1. Introduction The user of an artificial neural-network wants to know when the prediction of the model is accurate and trustworthy. When target ground truth is unavailable, as is usually the case, one must instead rely upon surrogate measures that correlate with accuracy and trustworthiness in a robust way. Uncertainty quantification aims to provide such measures. Recently there has been an intensive effort towards a better understanding of uncertainty of neural-network predictions [1 , 2] . To quantify this uncertainty in away that informs on the efficacy of the model, and to identify its sources, is of key significance in many applications of machine-learning algorithms using neural networks, from real-time predictions to active learning [3 , 4 , 5 , 6 , 7 , 8 , 9] . When the outputs of neural networks can be viewed as probability distributions over possible output values, certain distributional measures naturally capture the uncertainty of the network predictions. For instance, if the output distribution is sharply peaked, one might expect the prediction to be accurate. To which extent this expectation is borne out, depends not only on the model architecture and parameters, but also on the input data (for example whether it is from a domain the model has knowledge about) . Bayesian inference [10] provides a theoretical framework to reason about the conditional distribution of model parameters, and of the model output, given the available training data. More precisely, given a neural network with parameters θ, a prior p (θ), and a training dataset D = { (x1 , y 1 ) ,(x2 , y2 ) , ···} of pairs (input, target), Bayesian arguments determine a distribution over the neural-network parameters p (θ|D) [11] . This so-called posterior distribution tells us the probability of different model parameters given the training dataset. Using this posterior distribution for the parameters, a corresponding posterior distribution of the neural-network predictions, p (y|D, x) = Zθ p (y|θ, x)p (θ|D)dθ, (1) can be derived. The posterior predictive distribution in Eq. (1) is the marginalization over model parameters θ, conditioned on a particular input x that is either previously unseen or contained in the training dataset. Together, the posterior distribution of model parameters and the posterior predicti

FAQs

How does Bayesian inference quantify uncertainty in neural-network predictions?

It forms posterior distributions over model parameters given the training data and then derives a posterior predictive distribution over outputs for a given input. The entropies of these posteriors provide uncertainty measures.

What are aleatoric and epistemic uncertainties in this framework?

Epistemic uncertainty corresponds to uncertainty about model parameters and relates to the entropy of the posterior parameter distribution. Aleatoric uncertainty stems from randomness in the data generation process and is linked to the entropy of the posterior predictive distribution.

Why can’t prediction accuracy be explained using marginalized uncertainty distributions alone?

Prediction accuracy depends intricately on both epistemic and aleatoric uncertainties. The dependence also reflects model architecture and dataset properties, requiring analysis beyond marginalized measures.

Aug 3, 2026

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