Quantum Bayesian inversion and conditional distributions

This is a research talk given on Zoom at the Oxford ZX-Calculus Seminar on the 19th of July in 2021. Title: Quantum Bayesian inversion and conditional distributions Abstract: Bayes’ theorem can be formulated in two ways depending on the given data. Both statements can be expressed in terms of the existence of a morphism satisfying certain conditions. Based on this categorical description, one can transfer these concepts to the quantum setting by a fully faithful functor described by stochastic Gelfand duality. This is a fancy way of saying that conditional probabilities are equivalently described by positive maps between the algebras of observables. We will review these recent developments and we will provide two corresponding quantum analogues of Bayes’ theorem. A playlist of all my online research talks are available at https://www.youtube.com/playlist?list=PLSx1kJDjrLRT3cW1mnFfnmEvHycsv0iAA Potentially useful background material for this talk (not in any particular order): 1) My lectures on classical Bayes' theorem from a categorical perspective https://www.youtube.com/playlist?list=PLSx1kJDjrLRQksb7H9fqRE8GVMJdkX-4A 2) A shorter version of this talk was given at QPL 2021 and is available at https://www.youtube.com/watch?v=WAknCGuuBdw 3) Stochastic Gelfand duality in this setting is explained in Section 2 of https://arxiv.org/abs/1708.00091 4) This talk is based on https://arxiv.org/abs/2102.01529 and https://arxiv.org/abs/2005.03886, the latter of which is joint work with Benjamin P. Russo