A convergent scheme for the Bayesian filtering problem based on the Fokker--Planck equation and deep splitting
<|reference_start|>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 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.<|reference_end|>
@article{bågmark2024a,
    title={A convergent scheme for the Bayesian filtering problem based on the
  Fokker--Planck equation and deep splitting},
    author={Kasper B{aa}gmark, Adam Andersson, Stig Larsson, Filip Rydin},
    journal={arXiv preprint arXiv:2409.14585},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.14585},
    primaryClass={math.NA cs.NA math.PR stat.CO stat.ML}
}