A Variational Approach to Bayesian Phylogenetic Inference

TL;DR

This work introduces VBPI, a variational framework for Bayesian phylogenetic inference that leverages Subsplit Bayesian Networks (SBNs) to model tree topologies and a structured, shared amortization of branch lengths across topologies. By employing multi-sample ELBO objectives (IWAE-style) and advanced gradient estimators (VIMCO, RWS) along with two parameterizations for branch lengths (Split-based and Primary Subsplit Pair) and subsplit support estimation, VBPI achieves competitive posterior approximations to MCMC methods with fewer iterations. The approach extends to time-measured phylogenies by incorporating coalescent priors and node-height reparameterizations, demonstrating strong performance on real data (e.g., influenza, Dengue, HCV) and providing reliable marginal likelihood estimates via importance sampling. Overall, VBPI offers a scalable, flexible alternative to MCMC for complex phylogenetic models, with practical applicability to model comparison and demographic inference, while highlighting avenues for further improvement using richer variational families and dynamic subsplit support.

Abstract

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.

Figures (15)

• Figure 1: Subsplit Bayesian networks and a simple example for a leaf set of 4 taxa (represented by $A$, $B$, $C$, and $D$ respectively). ( Left): General subsplit Bayesian networks. The solid backbone represents the complete and binary tree network $\mathcal{B}_\mathcal{X}^\ast$. The dashed arrows represent the additional dependence for more expressive SBNs. ( Middle(left)): Examples of phylogenetic trees (rooted) that are hypothesized to model the evolutionary history of the taxa. ( Middle(right)): The corresponding node assignments for the trees. For ease of illustration, subsplit $(Y,Z)$ is represented as $\frac{Y}{Z}$ in the graph. The dashed gray subgraphs represent trivial splitting processes where the subsplits are deterministically assigned, and are used purely to complement the networks such that the overall network has a fixed structure. ( Right): The SBN for these examples, which is $\mathcal{B}_\mathcal{X}^\ast$ in this case.
• Figure 2: An illustration for Example \ref{['example:subsplit_eff']}. ( Left): A base tree $\tau_0$ with 3-taxon clade assignments for the tip nodes. ( Right): The local topologies that can be taken by the subtrees in $C_1$.
• Figure 3: SBNs for unrooted trees. ( Left): A simple four taxon unrooted tree. The equivalence class consists of five rooted trees, and each can be obtained by rooting on one of the edges $\{1,2,3,4,5\}$. ( Middle (left)): Two exemplary rooted trees in the equivalence class when rooting on edges $1$ and $3$. ( Middle (right)): The corresponding SBN assignments for the two rooted trees. ( Right): An SBN for the unrooted tree with unobserved root node $S_1$.
• Figure 4: The two-pass algorithm for computing the SBN probabilities of unrooted trees in \ref{['eq:sbn_unrooted_computation']}. ( Left): The conditional probability for the parent-child subsplit pair appeared in the message updating formulas. The triangles denote subtrees in a simplified form. The subtrees on the other side of node $k$ are ignored. ( Middle): The postorder (solid black) and preorder (dashed gray) traversals when node $k$ is chosen as the root node (denoted as a square node). ( Right): The root subsplit $(V\cup Z, X\cup Y)$ and parent-child subsplit pairs $(X,Y)|(V\cup Z, X\cup Y), (V,Z)|(V\cup Z, X\cup Y)$ for edge $e$.
• Figure 5: A comparison on four heuristic approaches for subsplit support estimation. ( Left): The sum of posterior probabilities of trees covered by SBNs with subsplit support estimated by different methods. ( Middle): The coverage percentages for root subsplits and subsplit pairs given by different methods. ( Right): The coverage efficiencies (the portion of root subsplits/subsplit pairs from the candidate trees that also appear in the ground truth collections) for root subsplits and subsplit pairs given by different methods. The letters R and S in parentheses represent the root subsplit and subsplit pair respectively. The error bars show one standard error based on 10 independent runs.

Theorems & Definitions (11)

• Definition 1: Subsplit
• Example 1
• Definition 2: Subsplit Bayesian Network
• Definition 3: Compatible Node Assignment
• Lemma 4
• Definition 5: Consistent Parameterization
• Proposition 6
• Definition 7: Equivalence Relation
• Proposition 8
• Proposition 9