  Likelihood-based methods are widely considered the best approaches for
reconstructing ancestral states. Although much effort has been made to study
properties of these methods, previous works often assume that both the tree
topology and edge lengths are known. In some scenarios the tree topology might
be reasonably well known for the taxa under study. When sequence length is much
smaller than the number of species, however, edge lengths are not likely to be
accurately estimated. We study the consistency of the maximum likelihood and
empirical Bayes estimators of ancestral state of discrete traits in such
settings under a star tree. We prove that the likelihood-based reconstruction
is consistent under symmetric models but can be inconsistent under
non-symmetric models. We show, however, that a simple consistent estimator for
the ancestral states is available under non-symmetric models. The results
illustrate that likelihood methods can unexpectedly have undesirable properties
as the number of sequences considered get very large. Broader implications of
the results are discussed.
