Joint Maximum Likelihood of Phylogeny and Ancestral States Is Not Consistent

Maximum likelihood estimation in phylogenetics requires a means of handling unknown ancestral states. Classical maximum likelihood averages over these unknown intermediate states, leading to provably consistent estimation of the topology and continuous model parameters. Recently, a computationally efficient approach has been proposed to jointly maximize over these unknown states and phylogenetic parameters. Although this method of joint maximum likelihood estimation can obtain estimates more quickly, its properties as an estimator are not yet clear. In this article, we show that this method of jointly estimating phylogenetic parameters along with ancestral states is not consistent in general. We find a sizeable region of parameter space that generates data on a four-taxon tree for which this joint method estimates the internal branch length to be exactly zero, even in the limit of infinite-length sequences. More generally, we show that this joint method only estimates branch lengths correctly on a set of measure zero. We show empirically that branch length estimates are systematically biased downward, even for short branches.