Consistency of the Maximum Likelihood Estimator of Evolutionary Tree

Maximum likelihood estimation (MLE) methods are widely used for evolutionary tree. As evolutionary tree is not a smooth parameter, the consistency of its MLE has been a topic of debate. It has been noted without proof that the classical proof of consistency by Wald holds for the MLE of evolutionary tree. Other proofs of consistency under various models were also proposed. Here we will discuss some shortcomings in some of these proofs and comment on the applicability of Wald's proof.