  In this PhD thesis, we explore and apply methods inspired by the free energy
principle to two important areas in machine learning and neuroscience. The free
energy principle is a general mathematical theory of the necessary
information-theoretic behaviours of systems that maintain a separation from
their environment. A core postulate of the theory is that complex systems can
be seen as performing variational Bayesian inference and minimizing an
information-theoretic quantity called the variational free energy. The thesis
is structured into three independent sections. Firstly, we focus on predictive
coding, a neurobiologically plausible process theory derived from the free
energy principle which argues that the primary function of the brain is to
minimize prediction errors, showing how predictive coding can be scaled up and
extended to be more biologically plausible, and elucidating its close links
with other methods such as Kalman Filtering. Secondly, we study active
inference, a neurobiologically grounded account of action through variational
message passing, and investigate how these methods can be scaled up to match
the performance of deep reinforcement learning methods. We additionally provide
a detailed mathematical understanding of the nature and origin of the
information-theoretic objectives that underlie exploratory behaviour. Finally,
we investigate biologically plausible methods of credit assignment in the
brain. We first demonstrate a close link between predictive coding and the
backpropagation of error algorithm. We go on to propose novel and simpler
algorithms which allow for backprop to be implemented in purely local,
biologically plausible computations.
