Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability
Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability
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分类概率中的可表示马尔可夫范畴及统计实验比较
DOI：
10.1016/j.tcs.2023.113896
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
E. F. Rischel
中科院分区：
文献类型：
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作者：
T. Fritz;Tomáš Gonda;Paolo Perrone;E. F. Rischel
马尔可夫类别是概率和统计数学基础的最新分类方法。在这里，通过陈述和证明二阶随机优势的等效条件来推进这种方法，二阶随机优势是一种广泛使用的通过分布来比较概率分布的方法。此外，我们通过陈述和证明经典的布莱克威尔-谢尔曼-斯坦定理，为比较马尔可夫范畴内的统计实验的理论奠定了基础。我们的版本不仅提供了对证明的新见解，而且其抽象性质也使结果更加普遍，自动专门针对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乱码的贝叶斯版本。在此过程中，我们定义并描述了可表示的马尔可夫类别，在这些类别中，人们可以在分布空间之间讨论马尔可夫核。我们通过探索概率单子的马尔可夫类别和克莱斯利类别之间的关系来做到这一点。
Markov categoriesare 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 characterizerepresentableMarkov 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.
DOI：
10.3842/sigma.2022.075
发表时间：
2022
期刊：
Symmetry, Integrability and Geometry: Methods and Applications
影响因子：
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作者：
M. Gerhold;S. Lachs;M. Schürmann
通讯作者：
M. Schürmann