Approximate Inference via Fibrations of Statistical Games
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We characterize a number of well known systems of approximate inference as loss models: lax sections of 2-fibrations of statistical games, constructed by attaching internally-defined loss functions to Bayesian lenses. Our examples include the relative entropy, which constitutes a strict section, and whose chain rule is formalized by the horizontal composition of the 2-fibration. In order to capture this compositional structure, we first introduce the notion of `copy-composition', alongside corresponding bicategories through which the composition of copy-discard categories factorizes. These bicategories are a variant of the 𝐂𝐨𝐩𝐚𝐫𝐚 construction, and so we additionally introduce coparameterized Bayesian lenses, proving that coparameterized Bayesian updates compose optically, as in the non-coparameterized case.
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