# On the convergence of the maximum likelihood estimator for the transition rate under a 2-state symmetric model

## TL;DR

This paper proves that the maximum likelihood estimator for the transition rate in a 2-state symmetric trait evolution model converges to the true value as the number of species increases, under certain regularity conditions.

## Contribution

It establishes the convergence and provides bounds for the MLE of the transition rate in a large-tree limit for a 2-state symmetric model, extending previous results to correlated trait data.

## Key findings

- MLE converges to the true transition rate under regularity conditions
- Provides an upper bound for the estimation error
- Results apply to various practical tree models such as Yule and coalescent trees

## Abstract

Maximum likelihood estimators are used extensively to estimate unknown parameters of stochastic trait evolution models on phylogenetic trees. Although the MLE has been proven to converge to the true value in the independent-sample case, we cannot appeal to this result because trait values of different species are correlated due to shared evolutionary history. In this paper, we consider a $2$-state symmetric model for a single binary trait and investigate the theoretical properties of the MLE for the transition rate in the large-tree limit. Here, the large-tree limit is a theoretical scenario where the number of taxa increases to infinity and we can observe the trait values for all species. Specifically, we prove that the MLE converges to the true value under some regularity conditions. These conditions ensure that the tree shape is not too irregular, and holds for many practical scenarios such as trees with bounded edges, trees generated from the Yule (pure birth) process, and trees generated from the coalescent point process. Our result also provides an upper bound for the distance between the MLE and the true value.