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The goal of branch length estimation in phylogenetic inference is to estimatethe divergence time between a set of sequences based on compositionaldifferences between them. A number of software is currently availablefacilitating branch lengths estimation for homogeneous and stationaryevolutionary models. Homogeneity of the evolutionary process imposes fixedrates of evolution throughout the tree. In complex data problems thisassumption is likely to put the results of the analyses in question.  In this work we propose an algorithm for parameter and branch lengthsinference in the discrete-time Markov processes on trees. This broad class ofnonhomogeneous models comprises the general Markov model and all its submodels,including both stationary and nonstationary models.  Here, we adapted the well-known Expectation-Maximization algorithm andpresent a detailed performance study of this approach for a selection ofnonhomogeneous evolutionary models. We conducted an extensive performanceassessment on multiple sequence alignments simulated under a variety ofsettings. We demonstrated high accuracy of the tool in parameter estimation andbranch lengths recovery, proving the method to be a valuable tool forphylogenetic inference in real life problems. $\empar$ is an open-source C++implementation of the methods introduced in this paper and is the first tooldesigned to handle nonhomogeneous data.

Empar: EM-based algorithm for parameter estimation of Markov models on trees