  Phylogenetic inference, grounded in molecular evolution models, is essential
for understanding the evolutionary relationships in biological data. Accounting
for the uncertainty of phylogenetic tree variables, which include tree
topologies and evolutionary distances on branches, is crucial for accurately
inferring species relationships from molecular data and tasks requiring
variable marginalization. Variational Bayesian methods are key to developing
scalable, practical models; however, it remains challenging to conduct
phylogenetic inference without restricting the combinatorially vast number of
possible tree topologies. In this work, we introduce a novel, fully
differentiable formulation of phylogenetic inference that leverages a unique
representation of topological distributions in continuous geometric spaces.
Through practical considerations on design spaces and control variates for
gradient estimations, our approach, GeoPhy, enables variational inference
without limiting the topological candidates. In experiments using real
benchmark datasets, GeoPhy significantly outperformed other approximate
Bayesian methods that considered whole topologies.
