A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics

Tobias Fritz

doi:10.1016/j.aim.2020.107239

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arXiv · 1908.07021

A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics

Tobias Fritz

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Abstract

We develop Markov categories as a framework for synthetic probability and statistics, following work of Golubtsov as well as Cho and Jacobs. This means that we treat the following concepts in purely abstract categorical terms: conditioning and disintegration; various versions of conditional independence and its standard properties; conditional products; almost surely; sufficient statistics; versions of theorems on sufficient statistics due to Fisher--Neyman, Basu, and Bahadur. Besides the conceptual clarity offered by our categorical setup, its main advantage is that it provides a uniform treatment of various types of probability theory, including discrete probability theory, measure-theoretic probability with general measurable spaces, Gaussian probability, stochastic processes of either of these kinds, and many others.

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Tobias Fritz. 2019-08-19. A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics. https://doi.org/10.1016/j.aim.2020.107239

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