Computing the joint distribution of the total tree length across loci in populations with variable size

TL;DR

This paper introduces a novel numerical method to compute the joint distribution of total tree lengths at two loci in populations with variable size, aiding demographic inference from genomic data.

Contribution

It presents the first method to compute these distributions for arbitrary sample sizes using hyperbolic PDEs and the method of characteristics.

Findings

Efficiently computes joint distributions of genealogical tree lengths.

Demonstrates accuracy and utility of the method in population genetics.

Extensible to multiple recombination events and structured populations.

Abstract

In recent years, a number of methods have been developed to infer complex demographic histories, especially historical population size changes, from genomic sequence data. Coalescent Hidden Markov Models have proven to be particularly useful for this type of inference. Due to the Markovian structure of these models, an essential building block is the joint distribution of local genealogical trees, or statistics of these genealogies, at two neighboring loci in populations of variable size. Here, we present a novel method to compute the marginal and the joint distribution of the total length of the genealogical trees at two loci separated by at most one recombination event for samples of arbitrary size. To our knowledge, no method to compute these distributions has been presented in the literature to date. We show that they can be obtained from the solution of certain hyperbolic systems of partial differential equations. We present a numerical algorithm, based on the method of characteristics, that can be used to efficiently and accurately solve these systems and compute the marginal and the joint distributions. We demonstrate its utility to study the properties of the joint distribution. Our flexible method can be straightforwardly extended to handle an arbitrary fixed number of recombination events, to include the distributions of other statistics of the genealogies as well, and can also be applied in structured populations.