Fast, accurate and simulation-free stochastic mapping

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Abstract
Mapping evolutionary trajectories of discrete traits onto phylogenies receives considerable attention in evolutionary biology. Given the trait observations at the tips of a phylogenetic tree, researchers are often interested where on the tree the trait changes its state and whether some changes are preferential in certain parts of the tree. In a model-based phylogenetic framework, such questions translate into characterizing probabilistic properties of evolutionary trajectories. Current methods of assessing these properties rely on computationally expensive simulations. In this paper, we present an efficient, simulation-free algorithm for computing two important and ubiquitous evolutionary trajectory properties. The first is the mean number of trait changes, where changes can be divided into classes of interest (e.g. synonymous/non-synonymous mutations). The mean evolutionary reward, accrued proportionally to the time a trait occupies each of its states, is the second property. To illustrate the usefulness of our results, we first employ our simulation-free stochastic mapping to execute a posterior predictive test of correlation between two evolutionary traits. We conclude by mapping synonymous and non-synonymous mutations onto branches of an HIV intrahost phylogenetic tree and comparing selection pressure on terminal and internal tree branches.