Compositional Active Inference II: Polynomial Dynamics. Approximate Inference Doctrines

TL;DR

This paper develops a compositional framework for active inference using polynomial dynamics, introducing approximate inference doctrines that connect statistical games with dynamical systems, with applications in computational neuroscience.

Contribution

It introduces a novel compositional theory of active inference based on polynomial functors and develops new approximate inference doctrines inspired by neuroscience.

Findings

Constructs monoidal bicategories of hierarchical inference systems

Develops two inference doctrines: Laplace and Hebb-Laplace

Demonstrates how these doctrines optimize Gaussian posteriors and parameters

Abstract

We develop the compositional theory of active inference by introducing activity, functorially relating statistical games to the dynamical systems which play them, using the new notion of approximate inference doctrine. In order to exhibit such functors, we first develop the necessary theory of dynamical systems, using a generalization of the language of polynomial functors to supply compositional interfaces of the required types: with the resulting polynomially indexed categories of coalgebras, we construct monoidal bicategories of differential and dynamical ``hierarchical inference systems'', in which approximate inference doctrines have semantics. We then describe ``externally parameterized'' statistical games, and use them to construct two approximate inference doctrines found in the computational neuroscience literature, which we call the `Laplace' and the `Hebb-Laplace' doctrines: the former produces dynamical systems which optimize the posteriors of Gaussian models; and the latter produces systems which additionally optimize the parameters (or `weights') which determine their predictions.