Unbiased Estimators Applied to the Ensemble Kalman-Bucy Filter

Abstract

Recent debiasing techniques are incorporated into the Ensemble Kalman-Bucy Filter (EnKBF). Specifically, a novel double randomization is applied. The EnKBF is a Monte Carlo (MC) method that approximates the Kalman-Bucy Filter (KBF), which in turn can be seen as the continuous-time version of the celebrated discrete-time Kalman Filter (KF). The KF is a method that combines sequential observations with an underlying dynamics model to predict the state of the quantity of interest. Our interest in the EnKBF comes from its relevance in high dimensions, where it overcomes the curse of dimensionality and outperforms other standard methods like the Particle Filter. We will consider debiasing techniques (also termed unbiased estimators) in order to improve the error-to-cost rate. Unbiased estimators are variance reduction techniques that produce unbiased and finite variance estimators. Applications of the EnKBF are numerous, from atmospheric sciences, numerical weather prediction, finance, machine learning, among others. Thus, improving the EnKBF is of interest. Numerical tests are done in order to evaluate the cost and the error-to-cost rate of the algorithm, where we consider Ornstein-Uhlenbeck processes. Specifically, a numerical comparison with the Multilevel Ensemble Kalman-Bucy Filter (MLEnKBF) is made using two different unbiased estimators, the coupled sum and the single term estimators. Additionally, we test two variants of the EnKBF, the Vanilla EnKBF, and the Deterministic EnKBF. We find that the error-to-cost rate is virtually the same, although the cost of the unbiased EnKBF is much higher.