A Convergent Scheme for the Bayesian Filtering Problem Based on the Fokker–Planck Equation and Deep Splitting

A numerical scheme for approximating the nonlinear filtering density is introduced and its convergence rate is established, theoretically under a parabolic Hörmander condition, and empirically for two examples. For the prediction step, between the noisy and partial measurements at discrete times, the scheme approximates the Fokker–Planck equation with a deep splitting scheme, and performs an exact update through Bayes' formula. This results in a classical prediction-update filtering algorithm that operates online for new observation sequences post-training. The algorithm employs a sampling-based Feynman–Kac approach, designed to mitigate the curse of dimensionality. Our convergence proof relies on the Malliavin integration-by-parts formula. As a corollary we obtain the convergence rate for the approximation of the Fokker–Planck equation alone, disconnected from the filtering problem.