  The marginal likelihood of a model is a key quantity for assessing the
evidence provided by the data in support of a model. The marginal likelihood is
the normalizing constant for the posterior density, obtained by integrating the
product of the likelihood and the prior with respect to model parameters. Thus,
the computational burden of computing the marginal likelihood scales with the
dimension of the parameter space. In phylogenetics, where we work with tree
topologies that are high-dimensional models, standard approaches to computing
marginal likelihoods are very slow. Here we study methods to quickly compute
the marginal likelihood of a single fixed tree topology. We benchmark the speed
and accuracy of 19 different methods to compute the marginal likelihood of
phylogenetic topologies on a suite of real datasets. These methods include
several new ones that we develop explicitly to solve this problem, as well as
existing algorithms that we apply to phylogenetic models for the first time.
Altogether, our results show that the accuracy of these methods varies widely,
and that accuracy does not necessarily correlate with computational burden. Our
newly developed methods are orders of magnitude faster than standard
approaches, and in some cases, their accuracy rivals the best established
estimators.
