DGM Is Dead; Long Live DGM-Star! A Quantization-Theoretic Approach to Gamma Rate Discretization

Modeling substitution rate heterogeneity as a discretized Gamma distribution is standard in phylogenetic inference. Ferretti et al. (2026) have recently demonstrated that the most commonly used equiprobable Gamma rate discretization, known simply as the discrete Gamma model (DGM), substantially overestimates average branch lengths in reconstructed phylogenies when the true rates are drawn from a continuous Gamma distribution. We propose an alternative method to discretize the Gamma rate distribution that considerably reduces the bias in estimated branch lengths and can be used as a drop-in replacement for the standard DGM. The method works especially well on large trees with short branches, yielding up to a 20-fold reduction in bias on simulated phylogenies. It is developed by applying quantization-theoretic tools to the analysis of a simplified star-phylogeny abstraction introduced by Ferretti et al. (2026), where the new method is unbiased by construction. We explain Ferretti et al.'s counterintuitive observation that increasing the number of rate categories can exacerbate the bias of the standard DGM, even though this should in principle make DGM a better approximation to the continuous case, and offer our method as an immediately available alternative.