Convergence of maximum likelihood supertree reconstruction

TL;DR

This paper analyzes the convergence rate of the maximum likelihood supertree method, proving it converges polynomially to the true species tree under certain conditions, including the exponential error model.

Contribution

It introduces an analytic approach to study convergence of the maximum likelihood supertree, including branch lengths, and verifies conditions for popular error models.

Findings

Convergence rate is polynomial under mild conditions.

Conditions verified for the exponential error model.

Method reconstructs both topology and branch lengths.

Abstract

Supertree methods are tree reconstruction techniques that combine several smaller gene trees (possibly on different sets of species) to build a larger species tree. The question of interest is whether the reconstructed supertree converges to the true species tree as the number of gene trees increases (that is, the consistency of supertree methods). In this paper, we are particularly interested in the convergence rate of the maximum likelihood supertree. Previous studies on the maximum likelihood supertree approach often formulate the question of interest as a discrete problem and focus on reconstructing the correct topology of the species tree. Aiming to reconstruct both the topology and the branch lengths of the species tree, we propose an analytic approach for analyzing the convergence of the maximum likelihood supertree method. Specifically, we consider each tree as one point of a metric space and prove that the distance between the maximum likelihood supertree and the species tree converges to zero at a polynomial rate under some mild conditions. We further verify these conditions for the popular exponential error model of gene trees.