Calibrated tree priors for relaxed phylogenetics and divergence time estimation

The use of fossil evidence to calibrate divergence time estimation has a long history. More recently, Bayesian Markov chain Monte Carlo has become the dominant method of divergence time estimation, and fossil evidence has been reinterpreted as the specification of prior distributions on the divergence times of calibration nodes. These so-called "soft calibrations" have become widely used but the statistical properties of calibrated tree priors in a Bayesian setting hashave not been carefully investigated. Here, we clarify that calibration densities, such as those defined in BEAST 1.5, do not represent the marginal prior distribution of the calibration node. We illustrate this with a number of analytical results on small trees. We also describe an alternative construction for a calibrated Yule prior on trees that allows direct specification of the marginal prior distribution of the calibrated divergence time, with or without the restriction of monophyly. This method requires the computation of the Yule prior conditional on the height of the divergence being calibrated. Unfortunately, a practical solution for multiple calibrations remains elusive. Our results suggest that direct estimation of the prior induced by specifying multiple calibration densities should be a prerequisite of any divergence time dating analysis.