Bayesian Filters via the Stochastic Category

Abstract

We extend the categorical treatment of Bayesian filters to models with feedback, whose transitions and observations may depend on the past observations. We work in the stochastic category, whose objects are measurable spaces and whose morphisms are Markov kernels, which we refer to as stochastic morphisms. We systematically parameterize stochastic morphisms by a measurable space, and develop the calculus of conditioning in a notation intuitively compatible with classical probability theory, requiring neither string diagrams nor densities. A stochastic morphism into a product space is regarded as a parameterized family of joint laws, and its conditional law is again a stochastic morphism, so that the rules of conditioning become equalities of stochastic morphisms. We define the Bayesian inverse with respect to a parameterized prior and prove Bayesian coherence. A model with feedback is then a model whose parameter space grows with the observations, and we derive the Bayesian filter, prediction and smoothing for such models, with standard Borel hidden states and countably generated observation spaces. The affine Gaussian stochastic morphisms parameterized by a measurable space form a family strictly larger than the Markov category $\mathbf{Gauss}$, closed under composition and Bayesian inversion, in which we obtain the Kalman filter and the Rauch-Tung-Striebel smoother with feedback.