Unbiased Estimation of the Vanilla and Deterministic Ensemble Kalman-Bucy Filters

TL;DR

This paper develops an unbiased estimator for the ensemble Kalman--Bucy filter in linear Gaussian settings, using randomization techniques to improve estimation accuracy and variance control, demonstrated on Ornstein-Uhlenbeck processes.

Contribution

It introduces a novel unbiased estimation method for the ensemble Kalman--Bucy filter, specifically for deterministic and vanilla variants, utilizing randomization in discretization and sample levels.

Findings

Unbiased estimators are effective in linear Gaussian filtering.

Comparison shows advantages over standard EnKBF and multilevel variants.

The method is demonstrated on Ornstein-Uhlenbeck processes across different dimensions.

Abstract

In this article we consider the development of an unbiased estimator for the ensemble Kalman--Bucy filter (EnKBF). The EnKBF is a continuous-time filtering methodology which can be viewed as a continuous-time analogue of the famous discrete-time ensemble Kalman filter. Our unbiased estimators will be motivated from recent work [Rhee \& Glynn 2010, [31]] which introduces randomization as a means to produce unbiased and finite variance estimators. The randomization enters through both the level of discretization, and through the number of samples at each level. Our estimator will be specific to linear and Gaussian settings, where we know that the EnKBF is consistent, in the particle limit $N \rightarrow \infty$, with the KBF. We highlight this for two particular variants of the EnKBF, i.e. the deterministic and vanilla variants, and demonstrate this on a linear Ornstein--Uhlenbeck process. We compare this with the EnKBF and the multilevel (MLEnKBF), for experiments with varying dimension size. We also provide a proof of the multilevel deterministic EnKBF, which provides a guideline for some of the unbiased methods.