Evidential Decision Theory via Partial Markov Categories

TL;DR

A synthetic Bayes theorem is proved; it is applied to define a syntactic partial theory of observations on any Markov categories whose normalisations can be computed in the original Markov category.

Abstract

We introduce partial Markov categories. In the same way that Markov categories encode stochastic processes, partial Markov categories encode stochastic processes with constraints, observations and updates. In particular, we prove a synthetic Bayes theorem and we apply it to define a syntactic partial theory of observations on any Markov category, whose normalisations can be computed in the original Markov category. Finally, we formalise Evidential Decision Theory in terms of partial Markov categories, and provide implemented examples.