To spike or not to spike: the whims of the Wonham filter in the strong noise regime

TL;DR

This paper analyzes the Shiryaev-Wonham filter for a two-state hidden Markov process under a weak observation-noise regime, revealing a trajectory-level spiking phenomenon and a smoothing transition when incorporating a finite window. It introduces a smoothing transform and a damping term to quantify smoothing, and proves a sharp phase-transition in the finite-horizon, trajectorial sense: for δ_γ ∼ C log γ/γ, the filter either converges to a spike process (fast feedback, C<2) or to the true hidden state (slow feedback, C>8), with discussion of the transition region. The work combines a decomposition of trajectories, a logit change of variables, path-transform techniques, and residual-control lemmas to bound damping effects and establish the limit behavior under strong noise, with implications for feedback-control systems using Wonham-type filters. The results deepen understanding of information content and misfires in filtering under strong observation noise and provide a framework for trajectory-wise analysis beyond invariant measures or L2 criteria.

Abstract

We study the celebrated Shiryaev-Wonham filter (1964) in its historical setup where the hidden Markov jump process has two states. We are interested in the weak noise regime for the observation equation. Interestingly, this becomes a strong noise regime for the filtering equations. Earlier results of the authors show the appearance of spikes in the filtered process, akin to a metastability phenomenon. This paper is aimed at understanding the smoothed optimal filter, which is relevant for any system with feedback. In particular, we exhibit a sharp phase transition between a spiking regime and a regime with perfect smoothing.

Figures (6)

• Figure 1.1: Numerical simulation of the hidden process ${\mathbf x}$ and the observation process ${\mathbf y}^\gamma$ for $\gamma = 10^2$. The challenge is to infer the drift of ${\mathbf y}^\gamma$, in spite of Brownian noise and in a very short window. Parameters are $\lambda=1.3$ and $p=0.4$. There are $10^6$ time steps to discretize $[0,10]$. The code is available at the online repository https://github.com/redachhaibi/Spikes-in-Classical-Filtering
• Figure 1.2: "The whims of the Wonham filter": Informally, on a very short time interval, it is difficult to distinguish between a change in the drift of ${\mathbf y}^\gamma$ and an exceptionnal time of Brownian motion. The figure shows a numerical simulation of the process $\left( \pi_t^\gamma \ ; \ t \geq 0 \right)$ for the same realization of ${\mathbf x}$ as Fig. \ref{['fig:processes_xy']}. Same time discretization. This time we chose the larger $\gamma=10^4$ to highlight spikes.
• Figure 1.3: Numerical simulation of the process $( \pi_t^{\delta, \gamma} \ ; \ t \geq 0)$ for the same realisation of ${\mathbf x}$ as Fig. \ref{['fig:processes_xy']}. Same time discretisation. We have $\gamma=10^4$ and $\delta_\gamma = C \frac{\log \gamma}{\gamma}$, with $C \in \{ \frac{1}{2} , 1, 2, 4, 8 \}$.
• Figure 2.1: Sketch of the two limiting processes. The graph $\mathcal{G} ({\mathbf x})$ of the hidden Markov pure jump process ${\mathbf x}$ is in red (solid lines), and the set-valued spike process ${\mathbb X}$ is the union of the blue graph (dashed lines) and red graph (solid lines).
• Figure 5.1: Decomposition of trajectory.

Theorems & Definitions (25)

• Remark 1.1: Duality between weak and strong noise
• Remark 2.1
• Theorem 2.2: Variant of the Main Theorem of bernardin2018spiking
• proof : Pointers to the proof
• Theorem 2.4: Main theorem
• proof : Sketch of proof
• Proposition 3.1
• proof
• Proposition 4.1
• proof