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We develop the compositional theory of active inference by introducingactivity, functorially relating statistical games to the dynamical systemswhich play them, using the new notion of approximate inference doctrine. Inorder to exhibit such functors, we first develop the necessary theory ofdynamical systems, using a generalization of the language of polynomialfunctors to supply compositional interfaces of the required types: with theresulting polynomially indexed categories of coalgebras, we construct monoidalbicategories 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 twoapproximate inference doctrines found in the computational neuroscienceliterature, which we call the `Laplace' and the `Hebb-Laplace' doctrines: theformer produces dynamical systems which optimize the posteriors of Gaussianmodels; and the latter produces systems which additionally optimize theparameters (or `weights') which determine their predictions.

Compositional Active Inference II: Polynomial Dynamics. Approximate Inference Doctrines