The SMC9 Is a Highly Accurate Approximation to the Ancestral Recombination Graph

Two sequentially Markov coalescent models (SMC and SMC9) are available as tractable approximations to the ancestral recombination graph (ARG). We present a Markov process describing coalescence at two fixed points along a pair of sequences evolving under the SMC9. Using our Markov process, we derive a number of new quantities related to the pairwise SMC9, thereby analytically quantifying for the first time the similarity between the SMC9 and the ARG. We use our process to show that the joint distribution of pairwise coalescence times at recombination sites under the SMC9 is the same as it is marginally under the ARG, which demonstrates that the SMC9 is, in a particular well-defined, intuitive sense, the most appropriate first-order sequentially Markov approximation to the ARG. Finally, we use these results to show that population size estimates under the pairwise SMC are asymptotically biased, while under the pairwise SMC9 they are approximately asymptotically unbiased.