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Phylogenetic and discrete-trait evolutionary inference depend heavily on anappropriate characterization of the underlying character substitution process.In this paper, we present random-effects substitution models that extend commoncontinuous-time Markov chain models into a richer class of processes capable ofcapturing a wider variety of substitution dynamics. As these random-effectssubstitution models often require many more parameters than their usualcounterparts, inference can be both statistically and computationallychallenging. Thus, we also propose an efficient approach to compute anapproximation to the gradient of the data likelihood with respect to allunknown substitution model parameters. We demonstrate that this approximategradient enables scaling of sampling-based inference, namely Bayesian inferencevia Hamiltonian Monte Carlo, under random-effects substitution models acrosslarge trees and state-spaces. Applied to a dataset of 583 SARS-CoV-2 sequences,an HKY model with random-effects shows strong signals of nonreversibility inthe substitution process, and posterior predictive model checks clearly showthat it is a more adequate model than a reversible model. When analyzing thepattern of phylogeographic spread of 1441 influenza A virus (H3N2) sequencesbetween 14 regions, a random-effects phylogeographic substitution model infersthat air travel volume adequately predicts almost all dispersal rates. Arandom-effects state-dependent substitution model reveals no evidence for aneffect of arboreality on the swimming mode in the tree frog subfamily Hylinae.Simulations reveal that random-effects substitution models can accommodate bothnegligible and radical departures from the underlying base substitution model.We show that our gradient-based inference approach is over an order ofmagnitude more time efficient than conventional approaches.

Random-effects substitution models for phylogenetics via scalable gradient approximations