Sequential Markov Chain Monte Carlo for Filtering of State-Space Models with Low or Degenerate Observation Noise

2025-11-07

AI Summary

This paper addresses discrete-time nonlinear filtering under severely degenerate or extremely low-observational-noise regimes—where observations depend solely on a hidden Markov state and the observation function exhibits structural degeneracy. We propose a manifold-based filtering framework that recursively updates the posterior density on a sequence of observation-induced constraint manifolds embedded in the state space. By rigorously characterizing these manifolds, we derive exact recursive update equations for the filter density and establish its convergence in the vanishing-noise limit. To enable practical implementation, we design a sequential Markov chain Monte Carlo (SMCMC) algorithm that performs efficient sampling and temporal approximation directly on the evolving manifolds. Unlike conventional methods relying on Gaussian approximations or restrictive degeneracy assumptions, our approach ensures both theoretical soundness and computational feasibility. Experiments on benchmark statistical physics and stochastic dynamical systems demonstrate substantial improvements in filtering accuracy and robustness over state-of-the-art techniques.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Security and Privacy: Large-scale security measurements

Abstract

We consider the discrete-time filtering problem in scenarios where the observation noise is degenerate or low. More precisely, one is given access to a discrete time observation sequence which at any time $k$ depends only on the state of an unobserved Markov chain. We specifically assume that the functional relationship between observations and hidden Markov chain has either degenerate or low noise. In this article, under suitable assumptions, we derive the filtering density and its recursions for this class of problems on a specific sequence of manifolds defined through the observation function. We then design sequential Markov chain Monte Carlo methods to approximate the filter serially in time. For a certain linear observation model, we show that using sequential Markov chain Monte Carlo for low noise will converge as the noise disappears to that of using sequential Markov chain Monte Carlo for degenerate noise. We illustrate the performance of our methodology on several challenging stochastic models deriving from Statistics and Applied Mathematics.

Problem

• Filtering state-space models with degenerate observation noise
• Developing sequential MCMC methods for low-noise scenarios
• Approximating filtering density on manifolds via observation functions

Innovation

• Sequential Markov Chain Monte Carlo for filtering
• Manifold-based filtering density recursion derivation
• Convergence analysis for degenerate and low noise