The bounded coalescent model: Conditioning a genealogy on a minimum root date

The coalescent model represents how individuals sampled from a population may have originated from a last common ancestor. The bounded coalescent model is obtained by conditioning the coalescent model such that the last common ancestor must have existed after a certain date. This conditioned model arises in a variety of applications, such as speciation, horizontal gene transfer or transmission analysis, and yet the bounded coalescent model has not been previously analysed in detail. Here we describe a new algorithm to simulate from this model directly, without resorting to rejection sampling. We show that this direct simulation algorithm is more computationally efficient than the rejection sampling approach. We also show how to calculate the probability of the last common ancestor occurring after a given date, which is required to compute the probability density of realisations under the bounded coalescent model. Our results are applicable in both the isochronous (when all samples have the same date) and heterochronous (where samples can have different dates) settings. We explore the effect of setting a bound on the date of the last common ancestor, and show that it affects a number of properties of the resulting phylogenies. All our methods are implemented in a new R package called BoundedCoalescent which is freely available online.