Computing coalescence rates for complex demographies and sampling configurations

Inference of population history from genetic data relies, implicitly or explicitly, on the distribution of coalescence times, because population size changes, migration, and admixture all leave characteristic signatures in genealogies. The distribution of pairwise coalescent rates in particular has emerged as a popular target for demographic inference methods. However, pairwise coalescent rates have limited power to resolve recent history, because recent coalescences in samples of size two are rare. In this article, we introduce demestats, a software library for computing first-coalescence and cross-coalescence rate functions for structured demographic models specified in the demes format. The method computes the instantaneous rate at which the first coalescence event occurs conditional on no prior coalescence for arbitrary sampling configurations, combines exact calculations with mean-field approximations for larger samples, and is differentiable with respect to model parameters. In simulations, these statistics recover recent population size change and recent migration more accurately than pairwise summaries. Applied to tree sequences inferred from the 1000 Genomes Project, we provide new insight into the rate of recent expansion in human populations.