Error Estimates for the Kernel Gain Function Approximation in the Feedback Particle Filter

TL;DR

This paper analyzes and improves a kernel-based algorithm for approximating the gain function in feedback particle filters, providing new theoretical insights, error analysis, and numerical validation for nonlinear non-Gaussian systems.

Contribution

It introduces new representations, algorithms, and an error analysis framework for the kernel-based gain function approximation in feedback particle filters.

Findings

Improved theoretical understanding of kernel-based gain approximation.

Derived asymptotic bias and variance estimates.

Validated results with numerical experiments.

Abstract

This paper is concerned with the analysis of the kernel-based algorithm for gain function approximation in the feedback particle filter. The exact gain function is the solution of a Poisson equation involving a probability-weighted Laplacian. The kernel-based method -- introduced in our prior work -- allows one to approximate this solution using {\em only} particles sampled from the probability distribution. This paper describes new representations and algorithms based on the kernel-based method. Theory surrounding the approximation is improved and a novel formula for the gain function approximation is derived. A procedure for carrying out error analysis of the approximation is introduced. Certain asymptotic estimates for bias and variance are derived for the general nonlinear non-Gaussian case. Comparison with the constant gain function approximation is provided. The results are illustrated with the aid of some numerical experiments.