New Preprint! | Areeb SM's website

We¹ have finished up a preprint (Fritz et al., 2025) for our work on empirical measures² and laws of large numbers³ in the context of Markov categories (woohoo!).

This project had been on the back burner for a while, in fact before I’d even started my PhD! Although of course, the approach we’re using was developed somewhat more recently :). In fact, it got so big that we needed a bunch of machinery on “partialising” Markov categories that ended up getting split off into its own article. But that’s a topic that’s got a fair bit of interest on its own accord.

We should be presenting it at this years Applied Category Theory conference in June. Hope to see plenty of people there!

Tobias Fritz, Tomáš Gonda, Antonio Lorenzin, Paolo Perrone, and I. ↩
Intuitively given by the relative frequencies of a sequence of “samples”. ↩
Or more generally, Glivenko-Cantelli theorems. ↩

References

Empirical Measures and Strong Laws of Large Numbers in Categorical Probability

Tobias Fritz, Tomáš Gonda, Antonio Lorenzin, Paolo Perrone, and Areeb Shah Mohammed

Mar 2025

Abs DOI arXiv

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.

References

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