Bayesian inversion wants to be a dagger, a way of reversing every channel that squares to the identity, but on the nose it is not, because reversal depends on the prior and only holds almost surely. Quotienting a category of probability spaces by almost-sure equality repairs this, and what emerges is a strikingly rigid, self-dual category.
The quotient
Start from a category whose objects are probability spaces and whose morphisms are channels between them, and identify two morphisms whenever they agree almost surely. For the quotient to be a category the relation must be a categorical congruence. Such a relation is an equivalence relation compatible with composition and with identities, so that composing representatives is well defined. The result is the influence category
What the quotient buys
Proposition on rigidity of the influence category
In
every object is self-dual, so the category is dagger-compact; the monoidal unit coincides with both the initial and the terminal object; the dagger is Bayesian inversion, now well defined because almost-sure equality absorbs the prior-dependence and the null-set ambiguity; and the whole category is equivalent to a category of input–output coupling diagrams. Inference reversed twice is inference, up to almost-sure equality.
This is the setting in which the dagger question raised for a categorical statistics has a clean answer. Inversion is a genuine involution once one works up to null sets.
References
- T. Fritz, A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics (Advances in Mathematics, 2020)