# Finding the best resolution for the Kingman–Tajima coalescent: theory and applications

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## Abstract

Many summary statistics currently used in population genetics and in phylogenetics depend only on a rather coarse resolution of the underlying tree (the number of extant lineages, for example). Hence, for computational purposes, working directly on these resolutions appears to be much more efficient. However, this approach seems to have been overlooked in the past. In this paper, we describe six different resolutions of the Kingman–Tajima coalescent together with the corresponding Markov chains, which are essential for inference methods. Two of the resolutions are the well-known \(n\)-coalescent and the lineage death process due to Kingman. Two other resolutions were mentioned by Kingman and Tajima, but never explicitly formalized. Another two resolutions are novel, and complete the picture of a multi-resolution coalescent. For all of them, we provide the forward and backward transition probabilities, the probability of visiting a given state as well as the probability of a given realization of the full Markov chain. We also provide a description of the state-space that highlights the computational gain obtained by working with lower-resolution objects. Finally, we give several examples of summary statistics that depend on a coarser resolution of Kingman’s coalescent, on which simulations are usually based.

### Keywords

\(n\)-Coalescent resolutions Tree shape statistics Computationally efficient and statistically sufficient inference### Mathematics Subject Classification (2000)

92D15 92D20 60J10## Notes

### Acknowledgments

We are grateful to Robert C. Griffiths for his insights, comments and guidance on this project, to John Rhodes and Mike Steel for their comments on an earlier version of this manuscript, and to Mike Steel for pointing out (Kemeny and Snell 1960, Defn. 6.3.1). We also thank the referee and the associate editor for their pertinent comments, particularly on the computational aspects of this work. During the initial course of this study, R.S. was supported by a research fellowship from the Royal Commission for the Exhibition of 1851 and T.S. was supported by a PhD scholarship of the German Science Foundation and a summer studentship of the Allan Wilson Centre. A.V. was supported by the ANR project MANEGE (ANR-09-BLAN-0215) and T.S. by the Swiss National Science foundation. R.S. and A.V. were supported in part by the chaire Modélisation Mathématique et Biodiversité of Veolia Environnement-École Polytechnique-Museum National d’Histoire Naturelle-Fondation X.

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