Metropolis-Hastings Sampling of Phylogenetic Networks: Correcting for Symmetries

2026-08-12

AI Summary

This study addresses the under-sampling of symmetric phylogenetic networks in Metropolis–Hastings sampling over leaf-labeled networks, which arises from neglecting internal symmetries. To correct this bias between leaf-labeled and fully labeled representations, the authors propose a symmetry-aware approach based on quotient Markov chains. A key innovation is the use of μ-vectors to efficiently determine the size of a network’s automorphism group. They prove that orchard networks—including trees—have trivial automorphism groups and thus require no correction. By integrating μ-vector computation, graph automorphism algorithms, and quotient chain construction, the method substantially accelerates automorphism group evaluation, enhancing both the accuracy and efficiency of posterior sampling. A Python implementation is provided to facilitate practical application.

Abstract

In phylogenetics, Metropolis-Hastings methods are commonly used to sample phylogenetic trees or networks, for example from Bayesian posteriors. These methods generally use transitions that distinguish all nodes involved, and thus require fully labelled representations of phylogenetic networks. We argue that sampling leaf-labelled phylogenetic networks demands a correction for the number of fully labelled representatives of a leaf-labelled network, or, equivalently, for its internal symmetry. Without correction, there is a danger of undersampling networks with internal symmetries. We show that this correction can be realized by a quotient construction on the Metropolis-Hastings Markov chain, which, in practice, requires the calculation of the size of the network's automorphism group. Using $μ$-vectors, we show that the automorphism group is trivial for orchard networks, and thus also for tree-child networks and trees. This implies that a correction for symmetry is not needed when sampling only from such network classes. More generally, using our Python implementation of the algorithms in this paper, we show that using $μ$-vectors can significantly speed up calculations of automorphism group sizes and thus of Metropolis-Hastings sampling of leaf-labelled networks.

Problem

• phylogenetic networks
• Metropolis-Hastings sampling
• symmetry correction
• automorphism group
• leaf-labelled networks

Innovation

• Metropolis-Hastings sampling
• phylogenetic networks
• symmetry correction
• automorphism group
• μ-vectors