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The coalescent of a sample from a binary branching process.

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  • 1Laboratoire de Probabilités, Statistique & Modélisation (LPSM), Sorbonne Université, CNRS, Paris, France; Center for Interdisciplinary Research in Biology (CIRB), Collège de France, CNRS, INSERM, PSL Research University, Paris, France.

Theoretical Population Biology
|April 29, 2018
PubMed
Summary

Sampling k tips from a branching process results in a coalescent point process (CPP). This k-sample tree can be modeled as a mixture of Bernoulli sampled CPPs, simplifying genealogical analysis.

Keywords:
Birth–death processCoalescent point processFinite exchangeable sequence.Incomplete samplingRandom treeSplitting tree

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Area of Science:

  • Stochastic processes
  • Probability theory
  • Mathematical biology

Background:

  • Branching processes model population dynamics with birth and death events.
  • Reduced trees from branching processes form coalescent point processes (CPPs).
  • Bernoulli sampling of CPPs also yields a CPP.

Purpose of the Study:

  • To characterize the genealogical structure of a k-sample from a branching process.
  • To establish the relationship between k-samples and Bernoulli sampled CPPs.
  • To provide a probabilistic framework for analyzing k-sample genealogies.

Main Methods:

  • Analysis of time-continuous binary branching processes.
  • Investigation of coalescent point processes (CPPs) and their properties.
  • Derivation of the distribution for k-samples from CPPs.

Main Results:

  • A k-sample from a branching process results in a tree that is a mixture of Bernoulli sampled CPPs.
  • The sampling probability in the mixture follows an explicit distribution.
  • The genealogy of a k-sample can be simulated by first drawing a sampling probability and then node depths.

Conclusions:

  • The k-sampling method provides a new perspective on analyzing population genealogies.
  • This framework simplifies the understanding of complex branching process dynamics.
  • The results offer a computationally tractable method for inferring evolutionary histories.