Exact decoding of the sequentially Markov coalescent

In statistical genetics, the sequentially Markov coalescent (SMC) is an important framework for approximating the distribution of genetic variation data under complex evolutionary models. Methods based on SMC are widely used in genetics and evolutionary biology, with significant applications to genotype phasing and imputation, recombination rate estimation, and inferring population history. SMC allows for likelihood-based inference using hidden Markov models (HMMs), where the latent variable represents a genealogy. Because genealogies are continuous, while HMMs are discrete, SMC requires discretizing the space of trees in a way that is complicated and can lead to bias. In this work, we propose a method that circumvents this requirement, enabling SMC-based inference to be performed in the natural setting of a continuous state space. We derive fast, exact methods for frequentist and Bayesian inference using SMC. Compared to existing methods, ours requires minimal user intervention or parameter tuning, no numerical optimization or E-M, and is faster and more accurate.