Inverses, disintegrations, and Bayesian inversion in quantum Markov categories

We analyze three successively more general notions of reversibility and statistical inference: ordinary inverses, disintegrations, and Bayesian inferences. We provide purely categorical definitions of these notions and show how each one is a strictly special instance of the latter in the cases of classical and quantum probability. This provides a categorical foundation for Bayesian inference as a generalization of reversing a process. To properly formulate these ideas, we develop quantum Markov categories by extending recent work of Cho–Jacobs and Fritz on classical Markov categories. We unify Cho–Jacobs’ categorical notion of almost everywhere (a.e.) equivalence in a way that is compatible with Parzygnat–Russo’s C∗-algebraic a.e. equivalence in quantum probability.