Goggin's corrected Kalman Filter: Guarantees and Filtering Regimes
Imon Banerjee, Itai Gurvich
TL;DR
This work revisits Goggin's corrected Kalman filter for non-Gaussian noises in a discrete-time linear state-space model and provides non-asymptotic, pre-limit MSE guarantees in terms of the regime-defining parameter $s_N$ and horizon $N$. By analyzing both non-centered and centered score-transform variants, the authors establish a $1/\sqrt{N}$ convergence rate in the balanced SNR regime $1/\sqrt{N} \lesssim s_N \lesssim \sqrt{N}$ and derive a batch-based Cramér--Rao lower bound that quantifies how close Goggin's filter can approach the optimum. They show that in the balanced regime Goggin's filter can outperform the Kalman filter, with a rigorous upper bound on the bias and MSE, and provide stronger results for the centered variant under dissipativity assumptions. The results advance practical non-Gaussian filtering and open avenues for extending these ideas to nonlinear filters and partially observed control problems.
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 Cramér-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.