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We elucidate the mathematical structure of Bayesian filtering, and Bayesianinference more broadly, by applying recent work on category theoreticalprobability, specifically the concept of a strongly representable Markovcategory. We show that filtering, along with related concepts such as conjugatepriors, arise from an adjunction: the process of taking a hidden Markov processis right adjoint to a forgetful functor. This has an interesting consequence.In practice, filtering is usually implemented using parametrised families ofdistributions. The Kalman filter is a particularly important example, whichuses Gaussians. Rather than calculating a new posterior each time, theimplementation only needs to udpate the parameters. This structure arisesnaturally from our adjunction; the correctness of such a model is witnessed bya map from the model into the system being modelled. Conjugate priors arisefrom this construction as a special case.  In showing this we define a notion of unifilar machine, which has its originsin the literature on epsilon-machines. Unifilar machines are useful as modelsof the "observable behaviour" of stochastic systems; we show additionally thatin the Kleisli category of the distribution monad there is a terminal unifilarmachine, and its elements are controlled stochastic processes, mappingsequences of the input alphabet probabilistically to sequences of the outputalphabet.

Unifilar Machines and the Adjoint Structure of Bayesian Filtering