This dissertation reports some first steps towards a compositional account of
active inference and the Bayesian brain. Specifically, we use the tools of
contemporary applied category theory to supply functorial semantics for
approximate inference. To do so, we define on the `syntactic' side the new
notion of Bayesian lens and show that Bayesian updating composes according to
the compositional lens pattern. Using Bayesian lenses, and inspired by
compositional game theory, we define fibrations of statistical games and
classify various problems of statistical inference as corresponding sections:
the chain rule of the relative entropy is formalized as a strict section, while
maximum likelihood estimation and the free energy give lax sections. In the
process, we introduce a new notion of `copy-composition'.
On the `semantic' side, we present a new formalization of general open
dynamical systems (particularly: deterministic, stochastic, and random; and
discrete- and continuous-time) as certain coalgebras of polynomial functors,
which we show collect into monoidal opindexed categories (or, alternatively,
into algebras for multicategories of generalized polynomial functors). We use
these opindexed categories to define monoidal bicategories of cilia: dynamical
systems which control lenses, and which supply the target for our functorial
semantics. Accordingly, we construct functors which explain the bidirectional
compositional structure of predictive coding neural circuits under the free
energy principle, thereby giving a formal mathematical underpinning to the
bidirectionality observed in the cortex. Along the way, we explain how to
compose rate-coded neural circuits using an algebra for a multicategory of
linear circuit diagrams, showing subsequently that this is subsumed by lenses
and polynomial functors.Comment: DPhil thesis; as submitted. Main change from v1: improved treatment
of statistical games. A number of errors also fixed, and some presentation
improved. Comments most welcom