Goggin's corrected Kalman Filter: Guarantees and Filtering Regimes

TL;DR

This paper analyzes Goggin's corrected Kalman Filter for non-Gaussian noise, providing explicit convergence guarantees across different noise regimes and extending understanding of filtering performance beyond Gaussian assumptions.

Contribution

It offers the first explicit convergence rate analysis of Goggin's filter in non-Gaussian settings, including degenerate and balanced regimes, without assuming Gaussian noise.

Findings

Goggin's filter converges at a quantifiable rate depending on observation noise levels.

Identifies regimes where trivial filters are nearly optimal.

Highlights the most effective regime for Goggin's filter performance.

Abstract

In this paper we revisit a non-linear filter for {\em non-Gaussian} noises that was introduced in [1]. Goggin proved that transforming the observations by the score function and then applying the Kalman Filter (KF) to the transformed observations results in an asymptotically optimal filter. In the current paper, we study the convergence rate of Goggin's filter in a pre-limit setting that allows us to study a range of signal-to-noise regimes which includes, as a special case, Goggin's setting. Our guarantees are explicit in the level of observation noise, and unlike most other works in filtering, we do not assume Gaussianity of the noises. Our proofs build on combining simple tools from two separate literature streams. One is a general posterior Cram\'er-Rao lower bound for filtering. The other is convergence-rate bounds in the Fisher information central limit theorem. Along the way, we also study filtering regimes for linear state-space models, characterizing clearly degenerate regimes -- where trivial filters are nearly optimal -- and a {\em balanced} regime, which is where Goggin's filter has the most value. \footnote{This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible.