  Variational Bayesian phylogenetic inference (VBPI) provides a promising
general variational framework for efficient estimation of phylogenetic
posteriors. However, the current diagonal Lognormal branch length approximation
would significantly restrict the quality of the approximating distributions. In
this paper, we propose a new type of VBPI, VBPI-NF, as a first step to empower
phylogenetic posterior estimation with deep learning techniques. By handling
the non-Euclidean branch length space of phylogenetic models with carefully
designed permutation equivariant transformations, VBPI-NF uses normalizing
flows to provide a rich family of flexible branch length distributions that
generalize across different tree topologies. We show that VBPI-NF significantly
improves upon the vanilla VBPI on a benchmark of challenging real data Bayesian
phylogenetic inference problems. Further investigation also reveals that the
structured parameterization in those permutation equivariant transformations
can provide additional amortization benefit.
