  Bayesian phylogenetic inference is currently done via Markov chain Monte
Carlo (MCMC) with simple proposal mechanisms. This hinders exploration
efficiency and often requires long runs to deliver accurate posterior
estimates. In this paper, we present an alternative approach: a variational
framework for Bayesian phylogenetic analysis. We propose combining subsplit
Bayesian networks, an expressive graphical model for tree topology
distributions, and a structured amortization of the branch lengths over tree
topologies for a suitable variational family of distributions. We train the
variational approximation via stochastic gradient ascent and adopt gradient
estimators for continuous and discrete variational parameters separately to
deal with the composite latent space of phylogenetic models. We show that our
variational approach provides competitive performance to MCMC, while requiring
much fewer (though more costly) iterations due to a more efficient exploration
mechanism enabled by variational inference. Experiments on a benchmark of
challenging real data Bayesian phylogenetic inference problems demonstrate the
effectiveness and efficiency of our methods.
