Conference and Talk

The present work develops a construction of a CD category of partial kernels from a particular type of Markov category called a partializable Markov category. These are a generalization of earlier models of categories of partial morphisms such as p-categories, dominical categories, restriction categories, etc. to a non-deterministic/non-cartesian setting. Here all morphisms are quasi-total, with a natural poset enrichment corresponding to one morphism being a restriction of the other. Furthermore, various properties important to categorical probability are preserved, such as positivity, representability, conditionals, Kolmogorov products, and splittings of idempotents. We additionally discuss an alternative notion of Kolmogorov product suitable for partial maps, as well as partial algebras for probability monads. The primary example is that of the partialization of the category of standard Borel spaces and Markov kernels. Other examples include variants where the distributions are finitely supported, or where one considers multivalued maps instead.

The Glivenko-Cantelli theorem is a uniform version of the strong law of large numbers. It states that for every IID sequence of random variables, the empirical measure converges to the underlying distribution (in the sense of uniform convergence of the CDF). In this work, we provide tools to study such limits of empirical measures in categorical probability. We propose two axioms, permutation invariance and empirical adequacy, that a morphism of type $X^\mathbb N \to X$ should satisfy to be interpretable as taking an infinite sequence as input and producing a sample from its empirical measure as output. Since not all sequences have a well-defined empirical measure, "such empirical sampling morphisms" live in quasi-Markov categories, which, unlike Markov categories, allow partial morphisms. Given an empirical sampling morphism and a few other properties, we prove representability as well as abstract versions of the de Finetti theorem, the Glivenko-Cantelli theorem and the strong law of large numbers. We provide several concrete constructions of empirical sampling morphisms as partially defined Markov kernels on standard Borel spaces. Instantiating our abstract results then recovers the standard Glivenko-Cantelli theorem and the strong law of large numbers for random variables with finite first moment. Our work thus provides a joint proof of these two theorems in conjunction with the de Finetti theorem from first principles.