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.

翻译：本文研究离散时间滤波问题，重点关注观测噪声退化或较低的场景。具体而言，给定离散时间观测序列，该序列在任意时刻$k$仅依赖于一个未被观测的马尔可夫链的状态。我们特别假设观测与隐马尔可夫链之间的函数关系具有退化或低噪声特性。在适当假设下，本文推导了此类问题在由观测函数定义的特定流形序列上的滤波密度及其递推公式。随后设计了序贯马尔可夫链蒙特卡洛方法以实现滤波器的时序串行近似。针对特定线性观测模型，我们证明了低噪声情形下序贯马尔可夫链蒙特卡洛方法在噪声消失时将收敛至退化噪声情形的对应方法。最后通过统计学与应用数学领域的若干挑战性随机模型验证了所提方法的性能。