Matrix-analytic Methods for the Evolution of Species Trees, Gene Trees, and Their Reconciliation

We consider the reconciliation problem, in which the task is to find a mapping of a gene tree into a species tree. In this paper we present a method, where for a given choice of parameters, we are able to compute the likelihood for alternative reconciliations. We describe a Markovian binary tree (MBT) model for the evolution of species trees, a quasi-birth-and-death (QBD) model for the evolution of gene trees, and provide a recursive algorithm to compute the likelihood of a given reconciliation between a species tree and a gene tree. We derive our results using the theory of matrix-analytic methods, prove them using rigorous mathematics together with decomposition of sample path arguments, and describe algorithms for the computation of a range of useful metrics. We illustrate the theory with examples and provide the physical interpretations of the discussed quantities, with a focus on the practical applications of the theory to incomplete data.