Exact distribution of divergence times from fossil ages and tree topologies

Being given a phylogenetic tree of both extant and extinct taxa in which the fossil ages are the only temporal information (namely, in which divergence times are considered unknown), we provide a method to compute the exact probability distribution of any divergence time of the tree with regard to any speciation (cladogenesis), extinction and fossilization rates under the Fossilized-Birth-Death model. We use this new method to obtain a probability distribution for the age of Amniota (the synapsid/sauropsid or bird/mammal divergence), one of the most-frequently used dating constraints. Our results suggest an older age (between about 322 and 340 Ma) than has been assumed by most studies that have used this constraint (which typically assumed a best estimate around 310-315 Ma) and provide, for the first time, a method to compute the shape of the probability density for this divergence time.