A convergent scheme for the Bayesian filtering problem based on the Fokker--Planck equation and deep splitting

2024/09/22 by Bågmark, Kasper, Andersson, Adam, Larsson, Stig +1 · 2 citations
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paper · doi:10.48550/arxiv.2409.14585

Abstract

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 in two numerical 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, combined with 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. The convergence proof relies on stochastic integration by parts from the Malliavin calculus. As a corollary we obtain the convergence rate for the approximation of the Fokker--Planck equation alone, disconnected from the filtering problem.

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• High-dimensional Bayesian filtering through deep density approximation
• Nonlinear filtering based on density approximation and deep BSDE prediction

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