The SMC’ is a highly accurate approximation to the ancestral recombination graph

Peter R. Wilton, Shai Carmi, Asger Hobolth
(Submitted on 12 Jan 2015)

Two sequentially Markov coalescent models (SMC and SMC’) are available as tractable approximations to the ancestral recombination graph (ARG). We present a model of coalescence at two fixed points along a pair of sequences evolving under the SMC’. Using our model, we derive a number of new quantities related to the pairwise SMC’, thereby analytically quantifying for the first time the similarity between the SMC’ and ARG. We use our model to show that the joint distribution of pairwise coalescence times at recombination sites under the SMC’ is the same as it is marginally under the ARG, demonstrating that the SMC’ is the canonical first-order sequentially Markov approximation to the pairwise ARG. Finally, we use these results to show that population size estimates under the pairwise SMC are asymptotically biased, while under the pairwise SMC’ they are approximately asymptotically unbiased.