We study the distribution of the unobserved states of two measure-valued diffusions of Fleming-Viot and Dawson-Watanabe type, conditional on observations from the underlying populations collected at past, present and future times. If seen as nonparametric hidden Markov models, this amounts to finding the smoothing distributions of these processes, which we show can be explicitly described in recursive form as finite mixtures of laws of Dirichlet and gamma random measures respectively. We characterize the time-dependent weights of these mixtures, accounting for potentially different time intervals between data collection times, and fully describe the implications of assuming a discrete or a nonatomic distribution for the underlying process that drives mutations. In particular, we show that with a nonatomic mutation offspring distribution, the inference automatically upweights mixture components that carry, as atoms, observed types shared at different collection times. The predictive distributions for further samples from the population conditional on the data are also identified and shown to be mixtures of generalized Polya urns, conditionally on a latent variable in the Dawson-Watanabe case.

翻译：我们研究Fleming-Viot和Dawson-Watanabe两种量值散射的未观测状态的分布情况,但以过去、现在和将来收集的基本人群的观察为条件。如果将这种分布视为非对称隐蔽的Markov模型,就等于找到这些过程的平滑分布,我们以递归形式分别明确描述为Drichlet定律和伽马随机测量法定律的有限混合物。我们将这些混合物的受时间限制的重量定性,考虑到数据收集时间之间的可能不同间隔,并充分描述假设离散或非原子分布对驱动突变的基本过程的影响。我们特别表明,在非原子突变后代分布的情况下,推断会自动增加混合物成分,这些成分在不同的收集时间里带有观察到的种类。根据数据从人群进一步采集样本的预测分布也被确定并显示为普通聚氨的混合物,条件是道尔-Watabe案的潜伏变量。