Goggin's corrected Kalman Filter: Guarantees and Filtering Regimes

Technical Summary: Goggin's Corrected Kalman Filter: Guarantees and Filtering Regimes

Problem Statement
The paper addresses the state estimation problem in linear state-space models where both the process noise ($w_t$) and observation noise ($v_t$) are non-Gaussian. In the standard Gaussian setting, the Kalman Filter (KF) is the Minimum Variance Unbiased Estimator (MVUE). However, when noise distributions are non-Gaussian, the KF is no longer optimal, though it remains the best linear unbiased estimator. The authors revisit a specific nonlinear filtering approach introduced by Goggin [1], which applies the KF to score-transformed observations. While Goggin established the asymptotic optimality of this "Goggin Filter" (GF) as a scaling parameter $N \to \infty$ in a specific regime, this paper seeks to provide non-asymptotic performance guarantees and characterize the filter's behavior across a broader spectrum of signal-to-noise ratios (SNR).

Methodology
The authors analyze the system under a generalized scaling framework where the state transition and observation equations depend on a parameter $N$ and a scaling factor $s_N$ for the observation noise. The core methodology involves:

1. Regime Partitioning: The parameter space is divided into three distinct filtering regimes based on the relationship between $s_N$ and $\sqrt{N}$:

   • Negligible SNR ($s_N \gg \sqrt{N}$): Observation noise dominates.
   • Large SNR ($s_N \ll 1/\sqrt{N}$): Observation noise is negligible.
   • Balanced SNR ($1/\sqrt{N} \lesssim s_N \lesssim \sqrt{N}$): The intermediate regime where the interaction between signal and noise is complex.

2. Theoretical Tools: The analysis bridges two literature streams:

   • Cramér–Rao Lower Bound (CRLB): Used to establish lower bounds on the Mean Squared Error (MSE) for sequential estimation.
   • Fisher Information Central Limit Theorem (CLT): Specifically, convergence rate bounds for the Fisher information of averages (based on Johnson [13] and related works). This allows the authors to approximate the information of non-Gaussian noise aggregates with the reciprocal of their variance, effectively treating aggregated signal noise as Gaussian for the purpose of lower-bounding the MSE.

3. Filter Variants: The paper analyzes two versions of the Goggin filter:

   • Non-centered GF: Applies the score function $\phi = -h'/h$ directly to the observation $Y_t$.
   • Centered GF: Applies the score function to the centered observation $(Y_t - \gamma \hat{x}_{t-1})$.

Key Contributions and Results

• Identification of Degenerate Regimes:

  • In the Negligible SNR regime, the authors prove that the trivial estimator $\hat{x}_t = E[X_t]$ is nearly optimal. Standard CRLB arguments are insufficient here due to looseness; instead, a Le Cam-type minimax analysis is used to show that no estimator can significantly outperform the prior mean.
  • In the Large SNR regime, the trivial estimator $\hat{x}_t = Y_t$ (using the observation directly) is nearly optimal.
  • In the Balanced SNR regime, both trivial estimators are strictly sub-optimal compared to the standard KF, establishing the necessity of more sophisticated filtering.

• Non-Asymptotic Guarantees for the Balanced Regime:

  • Theorem 1 (Non-centered GF): For $1 \ll s_N \lesssim \sqrt{N}$, the authors derive explicit bounds on the bias and variance of the non-centered Goggin filter. The MSE is shown to be $MSE^\star \pm O(1/s_N)$, where $MSE^\star$ is the best possible steady-state MSE. The relative sub-optimality is $O(\sqrt{N}/s_N^2)$.
  • Theorem 2 (Centered GF): Under stronger assumptions (strong dissipativity and linear growth of the score function), the centered Goggin filter achieves a tighter bound. Its MSE is $MSE^\star \pm O(\sqrt{s_N}/N^{3/4})$, resulting in a relative sub-optimality of $O(1/(\sqrt{s_N}N^{1/4}))$. This result justifies the numerical observations in [11] that centered transformations perform better, particularly when $s_N$ is small relative to $N^{1/4}$.

• Superiority over Kalman Filter:

  • Lemma 5 formally demonstrates that in the balanced regime (specifically $s_N \gg N^{1/4}$), the standard KF has an error that is $\Omega(MSE^\star)$, whereas the Goggin filter's error is $o(MSE^\star)$. This confirms that the GF provides a substantial improvement over the linear KF in non-Gaussian settings within this regime.

• Lower Bound Construction:

  • Theorem 3 establishes a tight lower bound on the best achievable MSE by analyzing the system in "batch" periods. By applying the Fisher Information CLT to the aggregated signal noise, the authors show that the lower bound behaves as if the noise were Gaussian, providing a benchmark against which the GF is measured.

Significance and Claims
The paper claims to strengthen Goggin's original asymptotic limit theorem by providing non-asymptotic (pre-limit) performance guarantees that are explicit in the SNR parameter $s_N$. By mapping the parameter space, the authors clarify exactly when the Goggin filter is valuable: it is most effective in the "balanced" SNR regime where simple filters fail and the standard KF is significantly sub-optimal.

The authors modestly note that their analysis does not cover the "Low SNR" window ($1/\sqrt{N} \lesssim s_N \lesssim 1$) where the first-order Taylor approximation central to the GF arguments becomes too loose. Furthermore, while the analysis relies on specific regularity assumptions (e.g., finite restricted Poincaré constants, smoothness of densities), the paper suggests that these tools (CRLB and Fisher Information CLT rates) could be extended to the control of Partially Observed Markov Decision Processes (MDPs), specifically to quantify how non-Gaussianity affects the separation principle in quadratic regulators. The work does not propose new experimental setups or applications beyond the theoretical characterization of these filtering regimes.