A Concise Mathematical Description of Active Inference in Discrete Time

TL;DR

This paper offers a clear, mathematically precise description of active inference in discrete time, including an example, detailed derivations, and Python code for implementation, aimed at clarifying the mathematical foundations.

Contribution

It provides a concise, rigorous mathematical formulation of active inference in discrete time, with detailed explanations and implementation code for the first time.

Findings

Provides a detailed example of action selection in active inference.

Includes Python code compatible with pymdp for practical implementation.

Clarifies mathematical details and derivations for better understanding.

Abstract

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.