A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics
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.
Cite
@article{arxiv.1908.07021,
title = {A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics},
author = {Tobias Fritz},
journal= {arXiv preprint arXiv:1908.07021},
year = {2020}
}
Comments
98 pages. v6: fixed error in Section 7. v7: incorporates referee's comments. v8: minor correction