Calculations for multi-type age-dependent binary branching processes

This article provides a method for calculating the joint probability density for the topology and the node times of a tree which has been produced by an multi-type age-dependent binary branching process and then sampled at a given time. These processes are a generalization, in two ways, of the constant rate birth-death process. There are a finite number of types of particle instead of a single type: each particle behaves in the same way as all others of the same type, but different types can behave differently. Secondly, the lifetime of a particle (before it either dies, changes to another type, or splits into 2) follows an arbitrary distribution, instead of the exponential lifetime in the constant rate case. Two applications concern models for macroevolution: the particles represent species, and the extant species are randomly sampled. In one application, 1-type and 2-type models for macroevolution are compared. The other is aimed at Bayesian phylogenetic analysis where the models considered here can provide a more realistic and more robust prior distribution over trees than is usually used. A third application is in the study of cell proliferation, where various types of cell can divide and differentiate.