Language and notation that is in line with standard texts on probability theory.

Detailed example of the T-maze environment of active inference.

Precise derivations of the inference and learning rules of active inference.

In this paper we present a concise mathematical description of active inference in discrete time. The main part of the paper serves as a basic introduction to the topic, including a detailed example of the action selection mechanism. The appendix discusses the more subtle mathematical details, targeting readers who have already studied the active inference literature but struggle to make sense of the mathematical details and derivations. Throughout, we emphasize precise and standard mathematical notation, ensuring consistency with existing texts and linking all equations to widely used references on active inference. Additionally, we provide Python code that implements the action selection and learning mechanisms described in this paper and is compatible with pymdp environments.

Active inference

Free energy principle

Bayesian inference

Tutorial

Mathematical review

© 2025 The Authors. Published by Elsevier Inc.