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

• URL: http://arxiv.org/abs/2409.14585v1
• Date: Sun, 22 Sep 2024 20:25:45 GMT
• Title: A convergent scheme for the Bayesian filtering problem based on the Fokker--Planck equation and deep splitting
• Authors: Kasper Bågmark, Adam Andersson, Stig Larsson, Filip Rydin,
• Abstract summary: A numerical scheme for approximating the nonlinear filtering density is introduced and its convergence rate is established. For the prediction step, 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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: A numerical scheme for approximating the nonlinear filtering density is introduced and its convergence rate is established, theoretically under a parabolic H\"{o}rmander 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.

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