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.

翻译：马尔可夫范畴是概率与统计数学基础的一种新范畴论方法。本文通过陈述并证明二阶随机占优（一种通过分布离散程度比较概率分布的广泛方法）的等价条件，推进了该方法的发展。此外，我们通过陈述并证明经典的布莱克威尔-谢尔曼-斯坦定理，为在马尔可夫范畴内比较统计实验的理论奠定基础。我们的版本不仅为证明提供了新见解，其抽象性还使结果更具普适性，能自动特化为测度论概率中的标准布莱克威尔-谢尔曼-斯坦定理，以及涉及先验依赖扰动的贝叶斯版本。在此过程中，我们定义并刻画了可表示的马尔可夫范畴，在其中可以讨论从分布空间出发或到达分布空间的马尔可夫核。我们通过探索马尔可夫范畴与概率单子克莱斯利范畴之间的关系来实现这一目标。