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：我们考虑对一类扩散过程演化的动态参数进行滤波的任务，该参数基于从共轭于扩散边际分布的似然函数中采集的离散时间数据，且存在一个定义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定义在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况，给出了通用结论。该情况通常导致滤波分布为由对偶过程状态空间索引的可数混合分布。我们以Cox-Ingersoll-Ross和Wright-Fisher扩散驱动的两个隐马尔可夫模型为例（其具有生灭型对偶过程），评估了多种近似策略的性能，并将其与基于死亡型对偶的可用精确策略以及扩散状态空间上的自助粒子滤波（作为通用基准）进行了比较。