Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability

Abstract

Markov categories are a recent categorical approach to the mathematical foundations of probability and statistics. Here, this approach is advanced by stating and proving equivalent conditions for second-order stochastic dominance, a widely used way of comparing probability distributions by their spread. Furthermore, we lay the foundation for the theory of comparing statistical experiments within Markov categories by stating and proving the classical Blackwell-Sherman-Stein Theorem. Our version not only offers new insight into the proof, but its abstract nature also makes the result more general, automatically specializing to the standard Blackwell-Sherman-Stein Theorem in measure-theoretic probability as well as a Bayesian version that involves prior-dependent garbling. Along the way, we define and characterize representable Markov categories, within which one can talk about Markov kernels to or from spaces of distributions. We do so by exploring the relation between Markov categories and Kleisli categories of probability monads.

De Finetti's Theorem in Categorical Probability

“…We now take a detour from the discussion of de Finetti's Theorem into the realm of Markov categories. All of the concepts defined in this section have been introduced in earlier works on categorical probability [7,17,19,18]. Nevertheless, in the interest of a self-contained presentation, we recall the main points here in a slightly less formal way, referring to the existing literature for full technical detail.…”

“…In BorelStoch, this amounts to the existence of regular conditional probabilities for measurable Markov kernels [18,Example 2.4].…”

“…For example, given an object X = {0, 1} in BorelStoch, we would like there be an object P X isomorphic to [0, 1] whose elements are themselves probability distributions over X. Indeed, BorelStoch allows for such a construction [18,Example 3.19]. However, this is not the case for FinStoch of course since P X cannot be a finite set.…”

“…Extending this requirement to morphisms with arbitrary domain A, which is expected to hold by the same reasoning, leads to the definition of distribution objects. Definition 3.8 (representable Markov category [18,Definition 3.7]). Given an object X in a Markov category C, a distribution object for X is an object P X together with natural bijections 4 C(A, X)…”

“…It differs from standard approaches by using more formal and abstract reasoning and it only depends on the particular measure-theoretic semantics insofar as the synthetic axioms may or may not be satisfied. A number of concepts and theorems of classical probability and statistics have been given a synthetic treatment in recent years [7,17,19,18]. We recall the ones relevant for de Finetti's Theorem in Section 3.…”