A Concise Mathematical Description of Active Inference in Discrete Time

Active inference is a theory that describes the behavior (action selection mechanism) of an agent in an environment. We aim to present a concise mathematical description of the theory so that a reader interested in the mathematical details can quickly find what they are looking for. We have paid special attention to choosing notation that is more in line with standard mathematical texts and is also descriptive, in the sense that dependencies are made explicit. The aim of this paper is not to justify the theory or convince the reader that this is right theory. The paper is divided into a main text and an appendix. The main text aims to present a clear and simple picture of active inference in discrete time that is accessible for people new to the topic. It is further subdivided into an inference part, which assumes the existence of a generative model, a learning part, in which we discuss how the agent can learn this model, and an example, illustrating the action selection mechanism. In the appendix the more subtle details and derivations are discussed. This part is aimed at people who have already studied the active inference literature but struggle to make sense of the mathematical details.