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Calculation of the log-likelihood stands as the computational bottleneck formany statistical phylogenetic algorithms. Even worse is its gradientevaluation, often used to target regions of high probability. Order ${\calO}\hspace{-0.2em}\left( N \right)$-dimensional gradient calculations based onthe standard pruning algorithm require ${\cal O}\hspace{-0.2em}\left( N^2\right)$ operations where N is the number of sampled molecular sequences. Withthe advent of high-throughput sequencing, recent phylogenetic studies haveanalyzed hundreds to thousands of sequences, with an apparent trend towardseven larger data sets as a result of advancing technology. Such large-scaleanalyses challenge phylogenetic reconstruction by requiring inference on largersets of process parameters to model the increasing data heterogeneity. To makethis tractable, we present a linear-time algorithm for ${\calO}\hspace{-0.2em}\left( N \right)$-dimensional gradient evaluation and apply itto general continuous-time Markov processes of sequence substitution on aphylogenetic tree without a need to assume either stationarity orreversibility. We apply this approach to learn the branch-specific evolutionaryrates of three pathogenic viruses: West Nile virus, Dengue virus and Lassavirus. Our proposed algorithm significantly improves inference efficiency witha 126- to 234-fold increase in maximum-likelihood optimization and a 16- to33-fold computational performance increase in a Bayesian framework.

Gradients do grow on trees: a linear-time ${\cal O}\hspace{-0.2em}\left( N \right)$-dimensional gradient for statistical phylogenetics