Calibrated Tree Priors for Relaxed Phylogenetics and Divergence Time Estimation

The use of fossil evidence to calibrate divergence time estimation has a long. More recently Bayesian MCMC has become the dominant method of time estimation and fossil evidence has been re-interpreted as the of prior distributions on the divergence times of calibration. These so-called "soft calibrations" have become widely used but the properties of calibrated tree priors in a Bayesian setting has not carefully investigated. Here we clarify that calibration densities, such those defined in BEAST 1. 5, do not represent the marginal prior distribution the calibration node. We illustrate this with a number of analytical results small trees. We also describe an alternative construction for a calibrated prior on trees that allows direct specification of the marginal prior of the calibrated divergence time, with or without the restriction monophyly. This method requires the computation of the Yule prior on the height of the divergence being calibrated. Unfortunately, a solution for multiple calibrations remains elusive. Our results that direct estimation of the prior induced by specifying multiple densities should be a prerequisite of any divergence time dating.