Empirical Bayes improvement of Kalman filter type of estimators

TL;DR

This paper introduces an empirical Bayes enhancement to Kalman filter estimators for mean estimation in Gaussian models, improving accuracy especially when the underlying state process is non-linear or non-Gaussian.

Contribution

It proposes a non-linear empirical Bayes method to improve Kalman filter estimates, applicable to both sequential and retrospective problems, under mild assumptions.

Findings

The method outperforms standard Kalman filter estimates in non-linear or non-Gaussian settings.

The improvement is strict when the state process deviates from linear Gaussian assumptions.

Applicable to both sequential and retrospective estimation scenarios.

Abstract

We consider the problem of estimating the means $\mu_i$ of $n$ random variables $Y_i \sim N(\mu_i,1)$, $i=1,\ldots ,n$. Assuming some structure on the $\mu$ process, e.g., a state space model, one may use a summary statistics for the contribution of the rest of the observations to the estimation of $\mu_i$. The most important example for this is the Kalman filter. We introduce a non-linear improvement of the standard weighted average of the given summary statistics and $Y_i$ itself, using empirical Bayes methods. The improvement is obtained under mild assumptions. It is strict when the process that governs the states $\mu_1,\ldots,\mu_n $ is not a linear Gaussian state-space model. We consider both the sequential and the retrospective estimation problems.