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arxiv: 2507.08386 · v2 · submitted 2025-07-11 · 🧬 q-bio.PE · stat.CO

Detecting Evolutionary Change-Points with Branch-Specific Substitution Models and Shrinkage Priors

Xiang Ji , Benjamin Redelings , Shuo Su , Hongcun Bao , Wu-Min Deng , Samuel L. Hong , Guy Baele , Philippe Lemey
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Marc A. Suchard
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Pith reviewed 2026-05-19 05:15 UTC · model grok-4.3

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keywords evolutionary change-pointsbranch-specific substitution modelsshrinkage priorsphylogenetic inferenceselection pressureBayesian methodsanalytical gradientsmpox evolution
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The pith

Combining branch-specific substitution models with shrinkage priors allows automatic identification of evolutionary change-points without prior knowledge of their locations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method to detect points where evolutionary substitution patterns change along branches of a phylogenetic tree. By applying shrinkage priors to the parameters of branch-specific models, the approach shrinks parameters on most branches toward a common value, highlighting only the true change-points. This eliminates the need for researchers to specify change-point locations in advance. An analytical gradient computation makes the method scalable, running in time linear with the number of branches. The authors demonstrate it on primate BRCA1 gene data for selection pressure and on mpox virus sequences for mutational dynamics, with major speed improvements over previous approaches.

Core claim

By integrating branch-specific substitution models with shrinkage priors, it is possible to automatically identify change-points in evolutionary dynamics on a phylogeny while simultaneously estimating distinct substitution parameters for each branch, enabled by a new analytical gradient algorithm whose computational time scales linearly with the number of parameters.

What carries the argument

Shrinkage priors on the branch-specific substitution parameters that automatically identify change-points by shrinking non-change branches to shared values, paired with an analytical gradient for efficient optimization.

Load-bearing premise

The shrinkage priors correctly distinguish true evolutionary change-points from random statistical noise in the data without missing real shifts or creating false ones.

What would settle it

A simulation study where known change-points are inserted into sequence data and the method fails to recover them accurately or the analytical gradient produces likelihood values differing from numerical checks.

Figures

Figures reproduced from arXiv: 2507.08386 by Benjamin Redelings, Guy Baele, Hongcun Bao, Marc A. Suchard, Philippe Lemey, Samuel L. Hong, Shuo Su, Wu-Min Deng, Xiang Ji.

Figure 1
Figure 1. Figure 1: Example of a 3-taxon tree. Sequence states [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Maximum clade credibility tree of the MPXV analysis. The dataset consists of 138 [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Maximum clade credibilty tree of the BRCA1 analysis. Numbers on branches represent [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Posterior sampling efficiency on all branch-specific substitution parameters for the [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
read the original abstract

Branch-specific substitution models are popular for detecting evolutionary change-points, such as shifts in selective pressure. However, applying such models typically requires prior knowledge of change-point locations on the phylogeny or faces scalability issues with large data sets. To address both limitations, we integrate branch-specific substitution models with shrinkage priors to automatically identify change-points without prior knowledge, while simultaneously estimating distinct substitution parameters for each branch. To enable tractable inference under this high-dimensional model, we develop an analytical gradient algorithm for the branch-specific substitution parameters where the computational time is linear in the number of parameters. We apply this gradient algorithm to infer selection pressure dynamics in the evolution of the BRCA1 gene in primates and mutational dynamics in viral sequences from the recent mpox epidemic. Our novel algorithm enhances inference efficiency, achieving up to a 126-fold speedup per iteration in maximum likelihood optimization when compared to central difference numerical gradient method and up to a 2026-fold improvement in computational performance within a Bayesian framework using Hamiltonian Monte Carlo sampler compared to conventional univariate random walk sampler.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes integrating branch-specific substitution models with shrinkage priors to enable automatic detection of evolutionary change-points on phylogenies without prior specification of their locations. It develops an analytical gradient algorithm for the branch-specific parameters whose per-iteration cost is stated to be linear in the number of parameters, and demonstrates the approach on BRCA1 selection dynamics in primates and mutational dynamics in mpox viral sequences, reporting up to 126-fold speedup versus central-difference gradients in ML optimization and up to 2026-fold improvement versus univariate random-walk sampling in HMC.

Significance. If the gradient derivation is exact and the shrinkage priors recover change-points with controlled bias and power, the method would address a practical scalability barrier in high-dimensional phylogenetic models and facilitate routine inference of selection or rate shifts on larger trees.

major comments (2)
  1. [Abstract and Results] Abstract and Results sections: the central claim of reliable automatic change-point detection is not supported by any reported quantitative validation (simulation recovery rates, false-positive rates under known shifts, or cross-validation performance). Only computational timings are supplied; this is load-bearing for the claim that the shrinkage prior correctly separates signal from noise.
  2. [Methods, analytical gradient derivation] Methods, analytical gradient derivation: the asserted linear scaling in the number of branch-specific parameters does not explicitly account for the additional tree traversals or partial-likelihood storage required once the Felsenstein pruning algorithm is applied to a fully branch-specific model; the shrinkage prior further couples all parameters, potentially introducing overhead not captured by the single-pass assumption.
minor comments (2)
  1. [Methods] The manuscript should state the precise form of the shrinkage prior (e.g., Laplace, horseshoe) and how its hyperparameters are set or inferred, since these are the only free parameters listed.
  2. [Figures and Tables] Figure legends and table captions should explicitly indicate whether reported speedups are wall-clock times, iteration counts, or effective sample sizes per unit time.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which help clarify the scope and presentation of our work. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract and Results] Abstract and Results sections: the central claim of reliable automatic change-point detection is not supported by any reported quantitative validation (simulation recovery rates, false-positive rates under known shifts, or cross-validation performance). Only computational timings are supplied; this is load-bearing for the claim that the shrinkage prior correctly separates signal from noise.

    Authors: We agree that the manuscript would be strengthened by quantitative validation of change-point recovery. The current results focus on real-data applications and computational performance, but we will add a dedicated simulation study reporting recovery rates, false-positive rates, and power under known shift scenarios. This addition will be included in the revised version. revision: yes

  2. Referee: [Methods, analytical gradient derivation] Methods, analytical gradient derivation: the asserted linear scaling in the number of branch-specific parameters does not explicitly account for the additional tree traversals or partial-likelihood storage required once the Felsenstein pruning algorithm is applied to a fully branch-specific model; the shrinkage prior further couples all parameters, potentially introducing overhead not captured by the single-pass assumption.

    Authors: The gradient algorithm performs a single forward-backward traversal to obtain all partial likelihoods and their derivatives simultaneously, so the dominant cost remains linear in the number of branch-specific parameters even under a fully branch-specific model. The shrinkage prior gradient is computed in an additional linear pass that does not require extra traversals. We will revise the Methods section to make these steps and the resulting complexity explicit. revision: partial

Circularity Check

0 steps flagged

No significant circularity; algorithmic construction is independent

full rationale

The paper develops a new analytical gradient for branch-specific substitution parameters and pairs it with shrinkage priors for change-point detection. Performance is benchmarked against external baselines (central-difference numerical gradients and univariate random-walk samplers) rather than being defined in terms of its own fitted outputs. No load-bearing step reduces by construction to a self-citation, a fitted parameter renamed as prediction, or an ansatz smuggled via prior work. The derivation chain is self-contained against external benchmarks and does not exhibit the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger records the minimal structural assumptions required by the described approach.

free parameters (1)
  • shrinkage prior hyperparameters
    Hyperparameters that control the strength of shrinkage toward a shared substitution process across branches; their values are not specified in the abstract.
axioms (1)
  • domain assumption The underlying continuous-time Markov substitution process on each branch is correctly specified by the chosen model family.
    Standard modeling assumption invoked when branch-specific parameters are introduced.

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Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    Al-Mohy, A. H. and Higham, N. J. 2011. Computing the action of the matrix exponential, with an application to exponential integrators. SIAM journal on scientific computing , 33(2): 488–511. ´Alvarez-Carretero, S., Kapli, P., and Yang, Z. 2023. Beginner’s guide on the use of paml to detect positive selection. Molecular biology and evolution , 40(4): msad041

    work page 2011

  2. [2]

    L., Cummings, M

    Ayres, D. L., Cummings, M. P., Baele, G., Darling, A. E., Lewis, P. O., Swofford, D. L., Huelsen- beck, J. P., Lemey, P., Rambaut, A., and Suchard, M. A. 2019. BEAGLE 3: Improved perfor- mance, scaling, and usability for a high-performance computing library for statistical phyloge- netics. Syst Biol., 68(6): 1052–1061

    work page 2019

  3. [3]

    A., and Alekseyenko, A

    Baele, G., Lemey, P., Bedford, T., Rambaut, A., Suchard, M. A., and Alekseyenko, A. V. 2012. Improving the accuracy of demographic and molecular clock model comparison while accommo- dating phylogenetic uncertainty. Molecular biology and evolution , 29(9): 2157–2167

    work page 2012

  4. [4]

    A., Bielejec, F., and Lemey, P

    Baele, G., Suchard, M. A., Bielejec, F., and Lemey, P. 2016. Bayesian codon substitution modelling to identify sources of pathogen evolutionary rate variation. Microbial Genomics , 2(6): e000057

    work page 2016

  5. [5]

    S., Bastide, P., Lemey, P., and Suchard, M

    Baele, G., Gill, M. S., Bastide, P., Lemey, P., and Suchard, M. A. 2021. Markov-modulated continuous-time markov chains to identify site-and branch-specific evolutionary variation in beast. Systematic biology , 70(1): 181–189

    work page 2021

  6. [6]

    J., Rambaut, A., and Suchard, M

    Drummond, A. J., Rambaut, A., and Suchard, M. A. 2025. BEAST X for Bayesian phylogenetic, phylogeographic and phylodynamic inference. Nature Methods

    work page 2025

  7. [7]

    and Gouy, M

    Boussau, B. and Gouy, M. 2006. Efficient likelihood computations with nonreversible models of evolution. Systematic biology , 55(5): 756–768

    work page 2006

  8. [8]

    M., Polson, N

    Carvalho, C. M., Polson, N. G., and Scott, J. G. 2010. The horseshoe estimator for sparse signals. Biometrika, 97(2): 465–480

    work page 2010

  9. [9]

    Cho, C. T. and Wenner, H. A. 1973. Monkeypox virus. Bacteriological reviews, 37(1): 1–18. 25 Dennis Jr, J. E. and Schnabel, R. B. 1996. Numerical methods for unconstrained optimization and nonlinear equations, volume 16. Siam

    work page 1973

  10. [10]

    E., Holbrook, A

    Didier, G., Glatt-Holtz, N. E., Holbrook, A. J., Magee, A. F., and Suchard, M. A. 2024. On the surprising effectiveness of a simple matrix exponential derivative approximation, with application to global sars-cov-2. Proceedings of the National Academy of Sciences , 121(3): e2318989121

    work page 2024

  11. [11]

    Felsenstein, J. 1973. Maximum likelihood and minimum-steps methods for estimating evolutionary trees from data on discrete characters. Syst Biol., 22(3): 240–249

    work page 1973

  12. [12]

    Felsenstein, J. 1981. Evolutionary trees from DNA sequences: a maximum likelihood approach. Journal of Molecular Evolution , 17: 368–376

    work page 1981

  13. [13]

    A., Ji, X., Zhang, Z., Lemey, P., and Suchard, M

    Fisher, A. A., Ji, X., Zhang, Z., Lemey, P., and Suchard, M. A. 2021. Relaxed random walks at scale. Systematic Biology , 70(2): 258–267

    work page 2021

  14. [14]

    A., Ji, X., Nishimura, A., Baele, G., Lemey, P., and Suchard, M

    Fisher, A. A., Ji, X., Nishimura, A., Baele, G., Lemey, P., and Suchard, M. A. 2023. Shrinkage-based random local clocks with scalable inference. Molecular biology and evolution , 40(11): msad242

    work page 2023

  15. [15]

    A., and Suchard, M

    Gangavarapu, K., Ji, X., Baele, G., Fourment, M., Lemey, P., Matsen IV, F. A., and Suchard, M. A. 2024. Many-core algorithms for high-dimensional gradients on phylogenetic trees.Bioinformatics, 40(2): btae030

    work page 2024

  16. [16]

    and Yang, Z

    Goldman, N. and Yang, Z. 1994. A codon-based model of nucleotide substitution for protein-coding dna sequences. Molecular biology and evolution , 11(5): 725–736

    work page 1994

  17. [17]

    Guindon, S., Dufayard, J.-F., Lefort, V., Anisimova, M., Hordijk, W., and Gascuel, O. 2010. New algorithms and methods to estimate maximum-likelihood phylogenies: assessing the performance of phyml 3.0. Systematic biology , 59(3): 307–321

    work page 2010

  18. [18]

    Hasegawa, M., Kishino, H., and Yano, T.-a. 1985. Dating of the human-ape splitting by a molecular clock of mitochondrial DNA. Journal of Molecular Evolution , 22(2): 160–174

    work page 1985

  19. [19]

    W., Magee, A

    Hassler, G. W., Magee, A. F., Zhang, Z., Baele, G., Lemey, P., Ji, X., Fourment, M., and Suchard, M. A. 2023. Data integration in bayesian phylogenetics. Annual review of statistics and its application, 10(1): 353–377. 26 H¨ ohna, S., Freyman, W. A., Nolen, Z., Huelsenbeck, J. P., May, M. R., and Moore, B. R. 2019. A bayesian approach for estimating branc...

    work page 2023

  20. [20]

    Holmes, E. C. 2009. The evolution and emergence of RNA viruses . Oxford University Press

    work page 2009

  21. [21]

    P., Larget, B., and Swofford, D

    Huelsenbeck, J. P., Larget, B., and Swofford, D. 2000. A compound Poisson process for relaxing the molecular clock. Genetics, 154(4): 1879–1892

    work page 2000

  22. [22]

    Ji, X., Griffing, A., and Thorne, J. L. 2016. A phylogenetic approach finds abundant interlocus gene conversion in yeast. Molecular biology and evolution , 33(9): 2469–2476

    work page 2016

  23. [23]

    Ji, X., Zhang, Z., Holbrook, A., Nishimura, A., Baele, G., Rambaut, A., Lemey, P., and Suchard, M. A. 2020. Gradients do grow on trees: a linear-time o (n)-dimensional gradient for statistical phylogenetics. Molecular biology and evolution , 37(10): 3047–3060

    work page 2020

  24. [24]

    A., Su, S., Thorne, J

    Ji, X., Fisher, A. A., Su, S., Thorne, J. L., Potter, B., Lemey, P., Baele, G., and Suchard, M. A. 2023. Scalable bayesian divergence time estimation with ratio transformations. Systematic Biology , 72(5): 1136–1153

    work page 2023

  25. [25]

    Lemey, P., Rambaut, A., and Pybus, O. G. 2006. Hiv evolutionary dynamics within and among hosts. Aids Rev, 8(3): 125–140

    work page 2006

  26. [26]

    O., Ji, X., Lemey, P., and Suchard, M

    Wertheim, J. O., Ji, X., Lemey, P., and Suchard, M. A. 2024. Random-effects substitution models for phylogenetics via scalable gradient approximations. Systematic Biology , 73(3): 562–578

    work page 2024

  27. [27]

    W., Rosenbluth, M

    Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E. 1953. Equation of State Calculations by Fast Computing Machines. The Journal of Chemical Physics , 21(6): 1087–1092

    work page 1953

  28. [28]

    Scheffler, K. 2013. Fubar: a fast, unconstrained bayesian approximation for inferring selection. Molecular biology and evolution , 30(5): 1196–1205. 27

    work page 2013

  29. [29]

    Muse, S. V. and Gaut, B. S. 1994. A likelihood approach for comparing synonymous and nonsyn- onymous nucleotide substitution rates, with application to the chloroplast genome. Molecular biology and evolution , 11(5): 715–724

    work page 1994

  30. [30]

    and Havel, T

    Najfeld, I. and Havel, T. F. 1995. Derivatives of the matrix exponential and their computation. Advances in applied mathematics , 16(3): 321–375

    work page 1995

  31. [31]

    Neal, R. M. 2011. MCMC using Hamiltonian dynamics. Handbook of Markov Chain Monte Carlo , 2(11). O’Toole, ´A., Neher, R. A., Ndodo, N., Borges, V., Gannon, B., Gomes, J. P., Groves, N., King, D. J., Maloney, D., Lemey, P., et al. 2023. Apobec3 deaminase editing in mpox virus as evidence for sustained human transmission since at least 2016. Science, 382(66...

    work page 2011

  32. [32]

    Kiem, C., and Bedford, T. 2024. Underdetected dispersal and extensive local transmission drove the 2022 mpox epidemic. Cell , 187(6): 1374–1386

    work page 2024

  33. [33]

    G., Scott, J

    Polson, N. G., Scott, J. G., and Windle, J. 2014. The bayesian bridge. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 76(4): 713–733

    work page 2014

  34. [34]

    Pond, S. L. K. and Frost, S. D. 2005. A genetic algorithm approach to detecting lineage-specific variation in selection pressure. Molecular biology and evolution , 22(3): 478–485

    work page 2005

  35. [35]

    Sherlock, C. 2021. Direct statistical inference for finite markov jump processes via the matrix exponential. Computational Statistics , 36(4): 2863–2887

    work page 2021

  36. [36]

    Stamatakis, A. 2014. Raxml version 8: a tool for phylogenetic analysis and post-analysis of large phylogenies. Bioinformatics, 30(9): 1312–1313

    work page 2014

  37. [37]

    A., Lemey, P., Baele, G., Ayres, D

    Suchard, M. A., Lemey, P., Baele, G., Ayres, D. L., Drummond, A. J., and Rambaut, A. 2018. Bayesian phylogenetic and phylodynamic data integration using BEAST 1.10. Virus Evol , 4(1): vey016

    work page 2018

  38. [38]

    and Nei, M

    Tamura, K. and Nei, M. 1993. Estimation of the number of nucleotide substitutions in the control region of mitochondrial dna in humans and chimpanzees. Molecular biology and evolution , 10(3): 512–526. 28

    work page 1993

  39. [39]

    L., Kishino, H., and Painter, I

    Thorne, J. L., Kishino, H., and Painter, I. S. 1998. Estimating the rate of evolution of the rate of molecular evolution. Mol Biol Evol. , 15(12): 1647–1657

    work page 1998

  40. [40]

    O., Leigh Brown, A

    Wertheim, J. O., Leigh Brown, A. J., Hepler, N. L., Mehta, S. R., Richman, D. D., Smith, D. M., and Kosakovsky Pond, S. L. 2014. The global transmission network of hiv-1. The Journal of infectious diseases, 209(2): 304–313

    work page 2014

  41. [41]

    Yang, Z. 1994. Maximum likelihood phylogenetic estimation from DNA sequences with variable rates over sites: approximate methods. Journal of Molecular Evolution , 39(3): 306–314

    work page 1994

  42. [42]

    Yang, Z. 1998. Likelihood ratio tests for detecting positive selection and application to primate lysozyme evolution. Molecular biology and evolution , 15(5): 568–573

    work page 1998

  43. [43]

    Yang, Z. 2007. Paml 4: phylogenetic analysis by maximum likelihood. Molecular biology and evolution, 24(8): 1586–1591

    work page 2007

  44. [44]

    and Nielsen, R

    Yang, Z. and Nielsen, R. 1998. Synonymous and nonsynonymous rate variation in nuclear genes of mammals. Journal of molecular evolution , 46: 409–418

    work page 1998

  45. [45]

    and Nielsen, R

    Yang, Z. and Nielsen, R. 2002. Codon-substitution models for detecting molecular adaptation at individual sites along specific lineages. Molecular biology and evolution , 19(6): 908–917. 29

    work page 2002