Mathematical foundations for a compositional account of the Bayesian brain

Thesis

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.

Because category theory is unfamiliar to many computational neuroscientists and cognitive scientists, we have made a particular effort to give clear, detailed, and approachable expositions of all the category-theoretic structures and results of which we make use. We hope that this dissertation will prove helpful in establishing a new "well-typed'' science of life and mind, and in facilitating interdisciplinary communication.

Authors: St Clere Smithe, TB

• DOI: 10.5287/ora-kzjqyop2d
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Bayes' law; relative entropy; fibration; statistical games; neural circuits; polynomial functor; chain rule; predictive coding; dynamical systems; functorial semantics; Bayesian; applied category theory; compositionality; Bayesian brain; cybernetics; copy-composition; category theory; open systems; free energy
• Subjects: Applied category theory; Dynamics; Categories (Mathematics); Computational neuroscience; Random dynamical systems; Bayesian statistics; Mathematical statistics; Markov processes; Mathematical neuroscience; Cybernetics