  Active inference is a state-of-the-art framework in neuroscience that offers
a unified theory of brain function. It is also proposed as a framework for
planning in AI. Unfortunately, the complex mathematics required to create new
models -- can impede application of active inference in neuroscience and AI
research. This paper addresses this problem by providing a complete
mathematical treatment of the active inference framework -- in discrete time
and state spaces -- and the derivation of the update equations for any new
model. We leverage the theoretical connection between active inference and
variational message passing as describe by John Winn and Christopher M. Bishop
in 2005. Since, variational message passing is a well-defined methodology for
deriving Bayesian belief update equations, this paper opens the door to
advanced generative models for active inference. We show that using a fully
factorized variational distribution simplifies the expected free energy -- that
furnishes priors over policies -- so that agents seek unambiguous states.
Finally, we consider future extensions that support deep tree searches for
sequential policy optimisation -- based upon structure learning and belief
propagation.
