# Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability

arXiv: https://arxiv.org/abs/2010.07416 · HTML: https://mlsanity.com/paper/2010.07416/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Oct 2020.

## Built on by (most cited first, 11 of 11)

- [Category Theory in Machine Learning](https://mlsanity.com/paper/2106.07032/) (2021)
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- [On Meta-Prompting](https://mlsanity.com/paper/2312.06562/) (2023)
- [Categorical semantics of compositional reinforcement learning](https://mlsanity.com/paper/2208.13687/) (2022)
- [A Categorical Semantics of Fuzzy Concepts in Conceptual Spaces](https://mlsanity.com/paper/2110.05985/) (2021)
- [Reinforcement Learning in Categorical Cybernetics](https://mlsanity.com/paper/2404.02688/) (2024)
- [Compositional Semantics for Probabilistic Programs with Exact Conditioning](https://mlsanity.com/paper/2101.11351/) (2021)
- [Reduce, Reuse, Recycle: Categories for Compositional Reinforcement Learning](https://mlsanity.com/paper/2408.13376/) (2024)
- [Supervised learning with probabilistic morphisms and kernel mean embeddings](https://mlsanity.com/paper/2305.06348/) (2023)
- [Transparent Semantic Spaces: A Categorical Approach to Explainable Word Embeddings](https://mlsanity.com/paper/2508.20701/) (2025)
- [Weaves, Wires, and Morphisms: Formalizing and Implementing the Algebra of Deep Learning](https://mlsanity.com/paper/2604.07242/) (2026)

## Abstract (arXiv)

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 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.