We consider a homogeneous birth-death process with three different sampling schemes. First, individuals can be sampled through time and included in a reconstructed phylogenetic tree. Second, they can be sampled through time and only recorded as a point ‘occurrence’ along a timeline. Third, extant individuals can be sampled and included in the reconstructed phylogenetic tree with a fixed probability. We further consider that sampled individuals can be removed or not from the process, upon sampling, with fixed probability. We derive the probability distribution of the population size at any time in the past conditional on the joint observation of a reconstructed phylogenetic tree and a record of occurrences not included in the tree. We also provide an algorithm to simulate ancestral population size trajectories given the observation of a reconstructed phylogenetic tree and occurrences. This distribution can be readily used to draw inferences about the ancestral population size in the field of epidemiology and macroevolution. In epidemiology, these results will allow data from epidemiological case count studies to be used in conjunction with molecular sequencing data (yielding reconstructed phylogenetic trees) to coherently estimate prevalence through time. In macroevolution, it will foster the joint examination of the fossil record and extant taxa to reconstruct past biodiversity.

我们考虑采用三种不同抽样方案的同质出生-死亡过程。首先，可以随时间对个体进行采样，并将其包含在重建的系统发育树中。其次，它们可以随时间进行采样，并且仅记录为时间线上的一个点“发生”。第三，可以对现存个体进行采样并以固定概率将其包含在重建的系统发育树中。我们进一步认为，抽样个体可以在抽样时以固定概率从过程中移除或不移除。我们根据对重建的系统发育树和未包含在树中的事件记录的联合观察，得出过去任何时间种群规模的概率分布。我们还提供了一种算法，根据重建的系统发育树和事件的观察来模拟祖先种群规模轨迹。这种分布可以很容易地用来推断流行病学和宏观进化领域的祖先种群规模。在流行病学中，这些结果将允许流行病学病例计数研究的数据与分子测序数据（产生重建的系统发育树）结合使用，以连贯地估计随时间变化的流行率。在宏观进化方面，它将促进对化石记录和现存类群的联合检查，以重建过去的生物多样性。