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Bayes' rule tells us how to invert a causal process in order to update ourbeliefs in light of new evidence. If the process is believed to have a complexcompositional structure, we may observe that the inversion of the whole can becomputed piecewise in terms of the component processes. We study the structureof this compositional rule, noting that it relates to the lens pattern infunctional programming. Working in a suitably general axiomatic presentation ofa category of Markov kernels, we see how we can think of Bayesian inversion asa particular instance of a state-dependent morphism in a fibred category. Wediscuss the compositional nature of this, formulated as a functor on theunderlying category and explore how this can used for a more type-drivenapproach to statistical inference.

The Compositional Structure of Bayesian Inference