Variational phylogenetic inference with products over bipartitions

TL;DR

This work addresses the challenge of Bayesian inference over ultrametric phylogenies without relying on MCMC by introducing VIPR, a variational framework with a novel density over tree space derived from single-linkage clustering of pairwise coalescent times. The method defines a scalable variational family where independent log-normal pairwise times $t^{\{u,v\}}$ induce a differentiable, closed-form density $q_{\phi}(\tau,\mathbf{t})$, enabling efficient gradient-based optimization of the ELBO. Through comparisons with BEAST and VBPI on diverse datasets, VIPR achieves competitive marginal log-likelihoods and ELBOs while using fewer gradient evaluations, and it scales empirically as $\mathcal{O}(N^2)$, making it practical for larger taxon sets. The framework also supports multiple gradient estimators (LOOR, reparameterization, VIMCO) and provides a foundation for future enhancements such as mixtures, flows, or relaxed-clock extensions. Overall, VIPR offers a differentiable, MCMC-free alternative for inferring time-measured phylogenies with uncertainty quantification and efficient computation.

Abstract

Bayesian phylogenetics requires accurate and efficient approximation of posterior distributions over trees. In this work, we develop a variational Bayesian approach for ultrametric phylogenetic trees. We present a novel variational family based on coalescent times of a single-linkage clustering and derive a closed-form density of the resulting distribution over trees. Unlike existing methods for ultrametric trees, our method performs inference over all of tree space, it does not require any Markov chain Monte Carlo subroutines, and our variational family is differentiable. Through experiments on benchmark genomic datasets and an application to SARS-CoV-2, we demonstrate that our method achieves competitive accuracy while requiring significantly fewer gradient evaluations than existing state-of-the-art techniques.

Figures (5)

• Figure 1: Diagram of the sampling process for VIPR. Two possible example matrices $\mathbf{T}$ could be drawn using $t^{\{u,v\}} \sim q_\phi^{\{u,v\}}$ and result in the same phylogenetic tree $(\tau,\mathbf{t}) \sim q_\phi$ after single-linkage clustering. Entries of $\mathbf{T}$ that trigger a coalescence event are shown in bold. The form of $q_\phi^{\{u,v\}}$ is defined by the practitioner, while the expression for $q_\phi$ depends upon $q_\phi^{\{u,v\}}$ and is defined in Equation (\ref{['eqn:q']}).
• Figure 2: Variational inference results for dataset DS1.left: Density estimation for tree lengths. center: Density estimation for tree log-likelihoods. Estimates are formed from 1,000 samples from the variational posterior of each VI method and 97,500 samples from the BEAST gold standard. right: Trace plot of estimated marginal log-likelihood vs. iteration number (i.e., parameter update). Marginal log-likelihood was estimated using 500 importance samples for VBPI and 50 importance samples for VIPR methods.
• Figure 3: Variational inference results for the COVID-19 dataset.left: Density estimation for tree lengths. center: Density estimation for tree log-likelihoods. right: Trace plot of estimated marginal log-likelihood vs. iteration number (i.e., parameter update). Density estimation was performed in the same way as in Figure \ref{['fig:DS1']}.
• Figure 4: Slope of the logarithm of seconds-per-iteration vs. the logarithm of the number of taxa.Each VI method was run for 1,000 iterations on subsets of the COVID-19 dataset. The y-axis corresponds to the computational complexity of the algorithm as a function of number of taxa (i.e., 1 corresponds to linear complexity, 2 corresponds to quadratic complexity, etc.)
• Figure 5: Trace plots for all datasets. Trace plot of estimated marginal log-likelihood vs. iteration number (i.e., parameter update number). Marginal log-likelihood was estimated using 500 importance samples for VBPI and 50 importance samples for VIPR methods.

Theorems & Definitions (1)

• Proposition 1