A Bayesian extension of phylogenetic generalized least squares (PGLS): incorporating uncertainty in the comparative study of trait relationships and evolutionary rates

Phylogenetic comparative methods use tree topology, branch lengths, and models of phenotypic change to take into account non-independence in statistical analysis. However, these methods normally assume that trees and models are known without error. Approaches relying on evolutionary regimes also assume specific distributions of character states across a tree, which often result from ancestral state reconstructions that are subject to uncertainty. Several methods have been proposed to deal with some of these sources of uncertainty, but approaches accounting for all of them are less common. Here we show how Bayesian statistics facilitates this task while relaxing the homogeneous rate assumption of the well-known phylogenetic generalized least squares (PGLS) framework.  This Bayesian formulation allows uncertainty about phylogeny, evolutionary regimes, or other statistical parameters to be taken into account for studies as simple as testing for coevolution in two traits or as complex as te...