Metropolis-Hastings Sampling of Phylogenetic Networks: Correcting for Symmetries

What the math gives to ML

The paper identifies a concrete failure mode in MCMC on objects represented with redundant labels: a leaf-labelled object with a large internal automorphism group has fewer distinct labelled representatives, so naive sampling over fully labelled states gives it the wrong stationary mass. The transferable asset is the quotient-chain construction, which replaces representation-space counting by an orbit-size or automorphism-group correction while preserving detailed balance. This can be applied to Bayesian neural architecture search or posterior sampling over networks whose hidden units, experts, channels, or latent components are exchangeable, where permutation-equivalent parameterizations otherwise distort the target distribution.