We consider the task of filtering a dynamic parameter evolving as a diffusion process, given data collected at discrete times from a likelihood which is conjugate to the marginal law of the diffusion, when a generic dual process on a discrete state space is available. Recently, it was shown that duality with respect to a death-like process implies that the filtering distributions are finite mixtures, making exact filtering and smoothing feasible through recursive algorithms with polynomial complexity in the number of observations. Here we provide general results for the case of duality between the diffusion and a regular jump continuous-time Markov chain on a discrete state space, which typically leads to filtering distribution given by countable mixtures indexed by the dual process state space. We investigate the performance of several approximation strategies on two hidden Markov models driven by Cox-Ingersoll-Ross and Wright-Fisher diffusions, which admit duals of birth-and-death type, and compare them with the available exact strategies based on death-type duals and with bootstrap particle filtering on the diffusion state space as a general benchmark.

翻译：我们考虑对作为扩散过程演化的动态参数进行滤波的任务，给定数据以离散时间从与扩散边缘分布共轭的似然函数中采集，且存在离散状态空间上的通用对偶过程。近期研究表明，与死亡型过程的对偶性意味着滤波分布为有限混合分布，从而可通过多项式复杂度的递归算法实现精确滤波和平滑。本文针对扩散过程与离散状态空间上的正则跳跃连续时间马尔可夫链之间的对偶情形，提出了通用结果——此类对偶通常导致由对偶过程状态空间索引的可数混合分布构成的滤波分布。我们基于Cox-Ingersoll-Ross和Wright-Fisher扩散驱动的两个隐马尔可夫模型（两者皆拥有生灭型对偶），研究多种近似策略的性能，并将其与基于死亡型对偶的精确策略及作为通用基准的扩散状态空间上的自举粒子滤波进行比较。