  Active inference is a normative principle underwriting perception, action,
planning, decision-making and learning in biological or artificial agents. From
its inception, its associated process theory has grown to incorporate complex
generative models, enabling simulation of a wide range of complex behaviours.
Due to successive developments in active inference, it is often difficult to
see how its underlying principle relates to process theories and practical
implementation. In this paper, we try to bridge this gap by providing a
complete mathematical synthesis of active inference on discrete state-space
models. This technical summary provides an overview of the theory, derives
neuronal dynamics from first principles and relates this dynamics to biological
processes. Furthermore, this paper provides a fundamental building block needed
to understand active inference for mixed generative models; allowing continuous
sensations to inform discrete representations. This paper may be used as
follows: to guide research towards outstanding challenges, a practical guide on
how to implement active inference to simulate experimental behaviour, or a
pointer towards various in-silico neurophysiological responses that may be used
to make empirical predictions.
