Efficient computation in the IM model

In this paper we analyze the isolation-with-migration model in a continuous-time Markov chain framework, and derive analytical expressions for the probability densities of gene tree topologies with an arbitrary number of lineages. We combine these densities with both nucleotide-substitution and infinite sites mutation models and derive probabilities for use in maximum likelihood estimation. We demonstrate how to apply lumpability of continuous-time Markov chains to achieve a significant reduction in the size of the state-space under consideration. We use matrix exponentiation and spectral decomposition to derive explicit expressions for the case of two diploid individuals in two populations, when the data is given as alignment columns. We implement these expressions in order to carry out a maximum likelihood analysis and provide a simulation study to examine the performance of our method in terms of our ability to recover true parameters. Finally, we show how the performance depends on the parameters in the model.