  Here we introduce a general class of multiple calibration birth-death tree
priors for use in Bayesian phylogenetic inference. All tree priors in this
class separate ancestral node heights into a set of "calibrated nodes" and
"uncalibrated nodes" such that the marginal distribution of the calibrated
nodes is user-specified whereas the density ratio of the birth-death prior is
retained for trees with equal values for the calibrated nodes. We describe two
formulations, one in which the calibration information informs the prior on
ranked tree topologies, through the (conditional) prior, and the other which
factorizes the prior on divergence times and ranked topologies, thus allowing
uniform, or any arbitrary prior distribution on ranked topologies. While the
first of these formulations has some attractive properties the algorithm we
present for computing its prior density is computationally intensive. On the
other hand, the second formulation is always computationally efficient. We
demonstrate the utility of the new class of multiple-calibration tree priors
using both small simulations and a real-world analysis and compare the results
to existing schemes. The two new calibrated tree priors described in this paper
offer greater flexibility and control of prior specification in calibrated
time-tree inference and divergence time dating, and will remove the need for
indirect approaches to the assessment of the combined effect of calibration
densities and tree process priors in Bayesian phylogenetic inference.
