Gaussianity and the Kalman Filter: A Simple Yet Complicated Relationship

TL;DR

This paper clarifies that the Kalman filter's optimality does not depend on Gaussian noise assumptions, correcting common misconceptions and emphasizing its broader applicability beyond Gaussian noise scenarios.

Contribution

It dispels misconceptions about the Gaussian requirement for Kalman filters and highlights their rigorous guarantees under minimal distribution assumptions.

Findings

Kalman filter provides optimal performance without Gaussian noise assumptions.

Misconceptions lead to unnecessary complexity in non-Gaussian scenarios.

Kalman filter's guarantees extend to nonlinear models under certain conditions.

Abstract

One of the most common misconceptions made about the Kalman filter when applied to linear systems is that it requires an assumption that all error and noise processes are Gaussian. This misconception has frequently led to the Kalman filter being dismissed in favor of complicated and/or purely heuristic approaches that are supposedly "more general" in that they can be applied to problems involving non-Gaussian noise. The fact is that the Kalman filter provides rigorous and optimal performance guarantees that do not rely on any distribution assumptions beyond mean and error covariance information. These guarantees even apply to use of the Kalman update formula when applied with nonlinear models, as long as its other required assumptions are satisfied. Here we discuss misconceptions about its generality that are often found and reinforced in the literature, especially outside the traditional fields of estimation and control.