  Ancestral maximum likelihood (AML) is a method that simultaneously
reconstructs a phylogenetic tree and ancestral sequences from extant data
(sequences at the leaves). The tree and ancestral sequences maximize the
probability of observing the given data under a Markov model of sequence
evolution, in which branch lengths are also optimized but constrained to take
the same value on any edge across all sequence sites. AML differs from the more
usual form of maximum likelihood (ML) in phylogenetics because ML averages over
all possible ancestral sequences. ML has long been known to be statistically
consistent -- that is, it converges on the correct tree with probability
approaching 1 as the sequence length grows. However, the statistical
consistency of AML has not been formally determined, despite informal remarks
in a literature that dates back 20 years. In this short note we prove a general
result that implies that AML is statistically inconsistent. In particular we
show that AML can `shrink' short edges in a tree, resulting in a tree that has
no internal resolution as the sequence length grows. Our results apply to any
number of taxa.
