Error Analysis of the Stochastic Linear Feedback Particle Filter
This work provides theoretical guarantees for the FPF algorithm, which is widely used in applications, but the analysis is limited to the linear Gaussian setting, making it an incremental contribution.
The paper analyzes the convergence and long-term stability of the feedback particle filter (FPF) for linear Gaussian systems, proving that the mean-field limit is well-defined and stable, and providing uniform-in-time mean-squared error estimates for finite particle systems.
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.