A unified multi-resolution coalescent: Markov lumpings of the Kingman-Tajima n-coalescent

In this paper, we formulate six different resolutions of a continuous-time approximation of the Wright-Fisher sample genealogical process. We derive Markov chains for the six different approximations in the spirit of J.F.C. Kingman. These Markov chains are essential for inference methods. One of the resolutions is the well-known n-coalescent due to Kingman. The second resolution was mentioned by Tajima, but never explicitly formalized. The other resolutions are novel, and embed the objects of Kingman and Tajima into a general framework via the theory of lumped Markov chains. We show that any sample genealogical Markov chain is amenable to Kingman’s n-coalescent approximation if it has the lineage death chain as its lumped Markov chain. We formulate a lumped n-coalescents graph that embodies multiple n-coalescent resolutions of the underlying sample genealogical process and leads to computationally efficient inference.