  We introduce quantum Markov categories as a structure that refines and
extends a synthetic approach to probability theory and information theory so
that it includes quantum probability and quantum information theory. In this
broader context, we analyze three successively more general notions of
reversibility and statistical inference: ordinary inverses, disintegrations,
and Bayesian inverses. We prove that each one is a strictly special instance of
the latter for certain subcategories, providing a categorical foundation for
Bayesian inversion as a generalization of reversing a process. We unify the
categorical and $C^*$-algebraic notions of almost everywhere (a.e.)
equivalence. As a consequence, we prove many results including a universal
no-broadcasting theorem for S-positive categories, a generalized Fisher--Neyman
factorization theorem for a.e. modular categories, a relationship between error
correcting codes and disintegrations, and the relationship between Bayesian
inversion and Umegaki's non-commutative sufficiency.
