OxCSML seminar Friday 12 May 2023

Department of Statistics, University of Oxford

Probability, symmetry, and entropy with Markov categories. By Paolo Perrone (Research Associate, Dept. of Computer Science, Oxford)

<p>Abstract: Markov categories are a modern framework designed to deal with uncertainty and probability in terms of category theory. Most conceptual aspects of probability theory can be described in this way: for example stochastic dependence and independence, conditioning, and conditional independence. More importantly, several theorems of probability and information theory have been recently stated, interpreted, and even proven, purely in terms of Markov categories. One of the main advantages of the categorical approach is the treatment of situations of symmetry: a number of universal constructions can be ported from algebra and geometry to probability. This gives further systematic understanding of the symmetries and their action, and further unifies these apparently distinct areas of mathematics. Among the results that we can present this way we have de Finetti's theorem and the ergodic decomposition theorem for dynamical systems. In addition, several concepts of classical information theory, such as the notions of entropy and mutual information, and the data processing inequalities that they satisfy, can be studied using Markov categories, and can be recovered from enriched categorical principles. Last but not least, the graphical formalism of Markov categories satisfies a d-separation criterion (Markov property), and so it contains a native theory of graphical models. Depending on time and interest, we explore one or more of these aspects.</p>

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OxCSML seminar Friday 12 May 2023

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OxCSML seminar Friday 12 May 2023

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