# Coalescence in the diffusion limit of a Bienayme-Galton-Watson branching process

## TL;DR

This paper develops a statistical framework for estimating the time since the most recent common ancestor in populations modeled by a supercritical Bienayme-Galton-Watson process, using limited information and diffusion approximations.

## Contribution

It introduces a novel method for estimating the MRCA time and initial population size based on minimal data and diffusion limits, extending previous analyses.

## Key findings

- Derived a probability formula for a sample descending from a single ancestor.
- Provided confidence intervals for initial population size based on observed data.
- Constructed a joint likelihood surface for estimating population size and MRCA time.

## Abstract

We consider the problem of estimating the elapsed time since the most recent common ancestor of a finite random sample drawn from a population which has evolved through a Bienayme-Galton-Watson branching process. More specifically, we are interested in the diffusion limit appropriate to a supercritical process in the near-critical limit evolving over a large number of time steps. Our approach differs from earlier analyses in that we assume the only known information is the mean and variance of the number of offspring per parent, the observed total population size at the time of sampling, and the size of the sample. We obtain a formula for the probability that a finite random sample of the population is descended from a single ancestor in the initial population, and derive a confidence interval for the initial population size in terms of the final population size and the time since initiating the process. We also determine a joint likelihood surface from which confidence regions can be determined for simultaneously estimating two parameters, (1) the population size at the time of the most recent common ancestor, and (2) the time elapsed since the existence of the most recent common ancestor.