Error Analysis of the Stochastic Linear Feedback Particle Filter

TL;DR

This paper analyzes the convergence and stability of the feedback particle filter in linear Gaussian settings, establishing conditions for long-term stability and error bounds, thus advancing understanding of its theoretical properties.

Contribution

It provides the first rigorous analysis of the mean-field limit, stability, and error estimates for the feedback particle filter in linear Gaussian models.

Findings

Mean-field limit is well-defined with a unique strong solution.

The mean-field process is stable with respect to initial conditions.

Finite-N system exhibits long-term stability with uniform mean-squared error bounds.

Abstract

This paper is concerned with the convergence and long-term stability analysis of the feedback particle filter (FPF) algorithm. The FPF is an interacting system of $N$ particles where the interaction is designed such that the empirical distribution of the particles approximates the posterior distribution. It is known that in the mean-field limit ($N=\infty$), the distribution of the particles is equal to the posterior distribution. However little is known about the convergence to the mean-field limit. In this paper, we consider the FPF algorithm for the linear Gaussian setting. In this setting, the algorithm is similar to the ensemble Kalman-Bucy filter algorithm. Although these algorithms have been numerically evaluated and widely used in applications, their convergence and long-term stability analysis remains an active area of research. In this paper, we show that, (i) the mean-field limit is well-defined with a unique strong solution; (ii) the mean-field process is stable with respect to the initial condition; (iii) we provide conditions such that the finite-$N$ system is long term stable and we obtain some mean-squared error estimates that are uniform in time.