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Compositional Bayesian Inference and Stochastic Combs

Dylan Braithwaite ( University of Strathclyde, Glasgow (Scotland) )
In categorical probability, the process of updating a prior distribution according to Bayes law can be axiomatised as a kind of 'weak inverse' to a Markov kernel. While these are often computed approximately by numerical methods in probabilistic programming, we are interested in analysing the structure of composite Bayesian inverses, by analogy with the way in which automatic differentiation may be computed recursively.
In this talk I will survey some settings in which Bayesian inversion can be considered as a functor out of a Markov category, paying particular attention to a case where a kernel, paired with its Bayesian inverse, is considered as a 'stochastic comb'. Combs provide a model of morphisms with 'holes' which can be substituted for other morphisms or combs. This gives rise to a number of composition operations, supported by an intuitive graphical notation extending string diagrams for monoidal categories. I will explain how operations used in the construction of statistical models can be succinctly depicted in terms of comb composition, and sketch a way that stochastic combs might be used as a semantics for a probabilistic programming language.



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