Phase transition on the convergence rate of parameter estimation under an Ornstein-Uhlenbeck diffusion on a tree

TL;DR

This paper investigates how the growth of evolutionary trees affects the accuracy of estimating parameters in Ornstein-Uhlenbeck processes, revealing a phase transition in convergence rates linked to the tree's growth.

Contribution

It establishes a phase transition in the convergence rate of parameter estimation for Ornstein-Uhlenbeck processes on trees, connecting it to the tree's growth and the reconstruction problem.

Findings

A phase transition in the convergence rate of the MLE for the mean parameter.

Loss of $\

A novel estimation method for covariance parameters achieving $\

Abstract

Diffusion processes on trees are commonly used in evolutionary biology to model the joint distribution of continuous traits, such as body mass, across species. Estimating the parameters of such processes from tip values presents challenges because of the intrinsic correlation between the observations produced by the shared evolutionary history, thus violating the standard independence assumption of large-sample theory. For instance Ho and An\'e \cite{HoAne13} recently proved that the mean (also known in this context as selection optimum) of an Ornstein-Uhlenbeck process on a tree cannot be estimated consistently from an increasing number of tip observations if the tree height is bounded. Here, using a fruitful connection to the so-called reconstruction problem in probability theory, we study the convergence rate of parameter estimation in the unbounded height case. For the mean of the process, we provide a necessary and sufficient condition for the consistency of the maximum likelihood estimator (MLE) and establish a phase transition on its convergence rate in terms of the growth of the tree. In particular we show that a loss of $\sqrt{n}$-consistency (i.e., the variance of the MLE becomes $\Omega(n^{-1})$, where $n$ is the number of tips) occurs when the tree growth is larger than a threshold related to the phase transition of the reconstruction problem. For the covariance parameters, we give a novel, efficient estimation method which achieves $\sqrt{n}$-consistency under natural assumptions on the tree.