  Binary trait data record the presence or absence of distinguishing traits in
individuals. We treat the problem of estimating ancestral trees with time depth
from binary trait data. Simple analysis of such data is problematic. Each
homology class of traits has a unique birth event on the tree, and the birth
event of a trait visible at the leaves is biased towards the leaves. We propose
a model-based analysis of such data, and present an MCMC algorithm that can
sample from the resulting posterior distribution. Our model is based on using a
birth-death process for the evolution of the elements of sets of traits. Our
analysis correctly accounts for the removal of singleton traits, which are
commonly discarded in real data sets. We illustrate Bayesian inference for two
binary-trait data sets which arise in historical linguistics. The Bayesian
approach allows for the incorporation of information from ancestral languages.
The marginal prior distribution of the root time is uniform. We present a
thorough analysis of the robustness of our results to model mispecification,
through analysis of predictive distributions for external data, and fitting
data simulated under alternative observation models. The reconstructed ages of
tree nodes are relatively robust, whilst posterior probabilities for topology
are not reliable.
