  Biological data objects often have both of the following features: (i) they
are functions rather than single numbers or vectors, and (ii) they are
correlated due to phylogenetic relationships. In this paper we give a flexible
statistical model for such data, by combining assumptions from phylogenetics
with Gaussian processes. We describe its use as a nonparametric Bayesian prior
distribution, both for prediction (placing posterior distributions on ancestral
functions) and model selection (comparing rates of evolution across a
phylogeny, or identifying the most likely phylogenies consistent with the
observed data). Our work is integrative, extending the popular phylogenetic
Brownian Motion and Ornstein-Uhlenbeck models to functional data and Bayesian
inference, and extending Gaussian Process regression to phylogenies. We provide
a brief illustration of the application of our method.
