An accurate sequentially Markov conditional sampling distribution for the coalescent with recombination.

The sequentially Markov coalescent is a simplified genealogical process that aims to capture the essential features of the full coalescent model with recombination, while being scalable in the number of loci. In this article, the sequentially Markov framework is applied to the conditional sampling distribution (CSD), which is at the core of many statistical tools for population genetic analyses. Briefly, the CSD describes the probability that an additionally sampled DNA sequence is of a certain type, given that a collection of sequences has already been observed. A hidden Markov model (HMM) formulation of the sequentially Markov CSD is developed here, yielding an algorithm with time complexity linear in both the number of loci and the number of haplotypes. This work provides a highly accurate, practical approximation to a recently introduced CSD derived from the diffusion process associated with the coalescent with recombination. It is empirically demonstrated that the improvement in accuracy of the new CSD over previously proposed HMM-based CSDs increases substantially with the number of loci. The framework presented here can be adopted in a wide range of applications in population genetics, including imputing missing sequence data, estimating recombination rates, and inferring human colonization history.