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REVIEW 3 major objections 5 minor 107 references

Infinite Mixture Models for Improved Modeling of Across-Site Evolutionary Variation

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Bayesian infinite mixture models—Dirichlet process mixtures, hierarchical Dirichlet processes, and infinite hidden Markov models—fit viral alignments better than standard models, and the best prior type varies by data set.

desk verdict Genuine implementation and honest discussion, but the model rankings rest on a marginal likelihood estimator the authors themselves flag as unreliable, so the quantitative claims need validation. read the letter →

arxiv 2412.06042 v1 pith:J5AELACZ submitted 2024-12-08 q-bio.PE stat.ME

Mandev S. Gill , Guy Baele , Marc A. Suchard , Philippe Lemey This is my paper · ORCID
classification q-bio.PEstat.ME MSC 62F1592D15
keywords infinitemixturemodelsDirichletprocessmixtureshierarchicalprocesseshiddenMarkovacross-siteevolutionaryvariationBayesianphylogeneticsmarginallikelihoodestimationdatasquashingMCMC
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that Bayesian infinite mixture models—models in which the number of hidden evolutionary classes is unbounded and inferred from the data—should replace fixed a priori partitions when modeling how evolution varies across alignment sites. It goes beyond the standard Dirichlet process mixture by adding two Bayesian nonparametric priors: a hierarchical Dirichlet process that pools information across predefined groups of sites, and an infinite hidden Markov model that lets the evolutionary class of one site depend on the class of its neighbor. Analyzing respiratory syncytial virus, hepatitis C subtype 4, and rabies virus data, the paper finds that the best infinite mixture beats the best standard model in every scenario, and that with one exception every infinite mixture beats every standard model. The winning prior type differs by data set, so the practical claim is that all three priors belong in the modeling toolkit. The framework is implemented in a widely used Bayesian phylogenetic software package and scales through data-squashing MCMC and parallel likelihood evaluation.

What carries the argument

The load-bearing objects are three Bayesian nonparametric priors on the site-specific evolutionary parameters $\theta_i=(\rho_i,Q_i)$—the overall substitution rate and the relative exchange-rate matrix for site $i$. A Dirichlet process (built from a stick-breaking representation or Chinese restaurant process) yields a discrete random measure whose atoms are the distinct evolutionary categories, making the number of categories $K$ a random variable with countably infinite support. A hierarchical Dirichlet process couples group-specific Dirichlet processes through a shared discrete base distribution, so groups such as codon positions share the same set of categories while assigning them different weights. An infinite hidden Markov model uses the same construction with an unbounded number of states and makes the category at site $i$ depend on the category at site $i-1$, capturing spatial correlation along the alignment. Posterior sampling couples a data-squashing MCMC scheme that jointly proposes category updates for blocks of similar sites with parallel likelihood evaluations across the $s\times c$ site-category grid, and all models are compared by estimating the log marginal likelihood with the adjusted stabilized harmonic mean estimator.

What would settle it

Re-estimate the log marginal likelihood for every model on all three data sets with generalized stepping-stone or path sampling and check whether the rankings survive—specifically, whether HKY+IHMM keeps its roughly 100-unit lead over HKY+HDP on RSV A, HKY+DP keeps its lead on HCV, and HKY+IHMM keeps its more than 1000-unit lead on rabies; if the ordering changes materially, the paper's comparative claim fails.

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Extended reading notes

Core claim

On its own terms, the paper's central discovery is that the three Bayesian nonparametric priors are not interchangeable: Dirichlet process mixtures, hierarchical Dirichlet processes, and infinite hidden Markov models each capture a different structure in the data, and each is the best-fitting choice in at least one of the six data-set and substitution-model combinations analyzed. The best infinite mixture outperforms the best standard model in all scenarios, and with the exception of the Dirichlet process paired with a GTR substitution model on rabies data, infinite mixtures always outperform standard models. The largest gains come from the infinite hidden Markov model, which beats the next-best model by roughly 100 log marginal likelihood units on respiratory syncytial virus data and by more than 1000 units on rabies data. These fit gains are not cosmetic: the appendix shows that different across-site variation models produce different maximum clade credibility trees, split frequencies, and posterior distributions over tree space. Under infinite mixtures, the simpler HKY substitution model often fits better than GTR, the reverse of what happens under standard models.

Load-bearing premise

The load-bearing premise is that the stabilized harmonic mean estimator gives comparably accurate log marginal likelihoods across all models compared; the paper itself notes that path sampling and stepping-stone estimators are more accurate and that MCMC convergence and mixing were inconsistent across replicates, so if harmonic mean estimates are biased differently by model, the claimed rankings could change.

Editorial extensions

If this is right

  • Practicing phylogeneticists can replace fixed partitions and discretized-gamma rate models with infinite mixtures that infer the number of evolutionary categories and the category of each site from the data, removing a major a priori modeling choice.
  • Because the best-fitting prior changes with data set and substitution model, model comparison for across-site variation should include all three priors rather than defaulting to Dirichlet process mixtures.
  • The infinite hidden Markov model's large wins on some alignments imply that adjacent sites often evolve under correlated categories that exchangeable mixtures cannot capture.
  • Choice of across-site variation model changes posterior phylogenetic inferences, including maximum clade credibility trees, split frequencies, and treespace occupation.
  • Data-squashing MCMC and parallel likelihood evaluation make these analyses computationally feasible on alignments with thousands of sites.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • We infer that the infinite hidden Markov model's very large margins on rabies virus data provide indirect evidence for spatial autocorrelation in evolutionary parameters along real alignments, a mechanism that could be tested by comparing IHMM against a fixed codon-partition model with within-partition spatial ordering on additional protein-coding data sets.
  • We infer that the qualitative conclusion—infinite mixtures beat standard models—is more secure than the fine-grained ordering of HDP, IHMM, and DP, because the harmonic mean estimator the paper uses is one the authors themselves describe as less reliable than path sampling and stepping-stone methods.
  • We infer a natural extension: define HDP groups by gene or genomic region in whole-genome data, and couple site-level HDPs across tree branches to model branch-specific and site-specific variation jointly, both directions the paper lists as future work but does not test.
  • We infer that combining IHMM's spatial dependence with decoupled clustering of substitution rates versus exchange-rate parameters is a concrete model variant that could outperform any prior considered here.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper develops Bayesian nonparametric infinite mixture models for across-site evolutionary variation, extending Dirichlet process mixtures to hierarchical Dirichlet processes with codon-position groups (HDP-Codon) and infinite hidden Markov models (IHMM). The models are implemented in BEAST X, using a data-squashing MCMC strategy and BEAGLE parallelization. The authors compare the fit of these models with standard approaches (No Variation, Gamma, Codon, Codon+Gamma) under HKY and GTR substitution models on three viral data sets (RSV A, HCV subtype 4, RABV), using log marginal likelihood estimates obtained from a stabilized harmonic mean estimator. They report that infinite mixture models generally outperform standard models, that IHMM and HDP can substantially outperform DP models, and that the best-performing infinite mixture prior varies across data sets and substitution models. They also compare posterior phylogenetic trees, split frequencies, and treespace heatmaps. The main quantitative conclusions rest on a single set of marginal likelihood estimates without reported Monte Carlo uncertainty, and the authors explicitly acknowledge limitations of the estimator and MCMC mixing.

Significance. If the empirical rankings are reliable, the paper makes a useful contribution: it introduces two underused Bayesian nonparametric priors for site-heterogeneity modeling, provides a working implementation in the widely used BEAST X platform, combines data-squashing MCMC with BEAGLE parallel likelihood evaluation, and makes XML input files publicly available. The comparison across three viral data sets and several standard models is a sensible first evaluation, and the appendix on posterior tree differences adds practical information. However, the central claim that 'different types of infinite mixture models emerge as the best choices in different scenarios' is only as strong as the marginal likelihood estimates, and the current estimator plus acknowledged mixing problems leave the quantitative rankings vulnerable. The significance is therefore conditional on strengthening the model-comparison evidence; the methodological framework itself is a solid basis for a revised manuscript.

major comments (3)
  1. [Section 3, Table A1, and Section 4] The paper's central ranking claims rest on single stabilized harmonic mean estimates of log marginal likelihood, with no replication or uncertainty quantification. The Discussion explicitly concedes that path sampling, stepping-stone, and generalized stepping-stone outperform this estimator, and that 'we have observed inconsistent performance in MCMC convergence and mixing across replicates.' Many of the qualitative conclusions depend on small margins: for RSV A with GTR, HDP-Codon beats IHMM by 4 log units and IHMM beats DP by 3 log units; for HCV with GTR, DP beats IHMM by 2 log units (Table A1). The stabilized harmonic mean is known to be biased and high-variance, especially under poor mixing, so these margins may not be robust. The authors should either provide replicated marginal likelihood estimates with standard errors, implement a more reliable estimator for HDP/IHMM, or demonstrate that the within-model Monte Carlo error is much smaller than the reported differences. Without this, the claim that different infinite mixture priors win in different scenarios is not established.
  2. [Section 3, empirical Bayes HKY base distribution] The authors set the HKY transition/transversion ratio base distribution hyperparameters using estimates from HKY + No Variation analyses of the same data: 'we take an empirical Bayes approach and adopt the estimated mean and ten times the estimated standard deviation of transition/transversion rate estimates from HKY + No Variation models.' This uses the data twice and is asymmetric: HKY infinite mixture models receive data-informed priors, while GTR infinite mixture models and standard HKY models do not. Since the paper's conclusion that HKY-based mixtures outperform GTR-based mixtures is central to its 'best model varies' narrative, this prior specification may systematically favor HKY mixtures. The authors should either infer these hyperparameters jointly, use a fully hierarchical prior, or at a minimum assess sensitivity of the rankings to this empirical Bayes choice.
  3. [Section 2.4 and Section 4, MCMC convergence] The paper provides no convergence diagnostics, effective sample sizes, or replicate-level summaries for the MCMC runs, despite relying on an approximate data-squashing sampler and acknowledging inconsistent mixing. This matters not only for marginal likelihoods but also for the posterior quantities summarized in Table 1 and Figures A16-A20, including posterior medians, credibility intervals, and site-specific posterior means. The authors should report convergence diagnostics (e.g., ESS, Gelman-Rubin across replicates) for at least the main analyses, or state explicitly which results come from unreplicated single chains and how the acknowledged mixing problems affect those estimates.
minor comments (5)
  1. [Section 3.1 and Table A1] The model is referred to as 'HKY + HDP' in the text but 'HKY + HDP-Codon' in Table A1 and elsewhere; please standardize the naming throughout.
  2. [Appendix A.3] The claim of 'substantial differences' in phylogenetic posterior distributions should be reconciled with the reported split-frequency correlations of 0.97-1.00 and ASDSF values below 0.01; the heatmaps may show differences, but the split-frequency evidence suggests mostly similar posteriors, so the language should be tempered or the basis for 'substantial' should be clarified.
  3. [Section 2.4] The notation in the data-squashing description, such as 'n(t)zj * P(Xj|...)' and the subscript on zj, is confusing; using a clearer index for the category (e.g., k) and defining the star would improve readability.
  4. [References] There are several typographical errors in the reference list, including 'Jounal of Molecular Evolution', 'annd', 'orocess', 'Deparment', 'nucelotide', and 'Bayess'; a careful copyedit is needed.
  5. [Figure 1] In the RABV panel, the bar for HKY + IHMM is so much larger than the others that it compresses the visual differences among the remaining models; consider using a broken axis or a separate inset for the large values.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper is an empirical model comparison whose claims rest on independent likelihood and marginal-likelihood computations, not on fitted quantities renamed as predictions.

full rationale

The paper's central contribution is a Bayesian nonparametric framework (DP, HDP, IHMM) implemented in BEAST X and evaluated by log marginal likelihood on three viral data sets. The comparison is empirical, and the models are distinct priors over site-to-category assignments; none of the reported 'best model' conclusions is defined as the output of a fitted parameter, nor does any equation equate the prediction with its input. The only data-dependent prior specification is for HKY transition/transversion base distributions: 'we take an empirical Bayes approach and adopt the estimated mean and ten times the estimated standard deviation of transition/transversion rate estimates from HKY + No Variation models.' This is a modeling choice that makes the HKY mixture priors data-dependent, but it is not a derivation step, it is shared across the HKY DP/HDP/IHMM comparisons, and it does not force any particular ranking or define any predicted quantity. Self-citations (e.g., Redelings and Suchard 2005 for the harmonic mean estimator, Gill et al. 2013 for the skygrid prior) are standard methodological references and are not load-bearing in the sense of importing an unverified uniqueness theorem or ansatz. Concerns about the stabilized harmonic mean estimator's known bias and the paper's own report of 'inconsistent performance in MCMC convergence and mixing across replicates' bear on the reliability of the model ranking, not on circularity of the derivation; they are correctness risks, not circularity.

Assumptions & free parameters 6 free parameters · 11 assumptions · 0 invented entities

The framework leans on established Bayesian nonparametric constructions (stick-breaking, Polya urn, HDP, HDP-HMM), standard phylogenetic likelihood computations, and a set of prior and grouping choices specific to these analyses. The main ad hoc element is the empirical Bayes setting of HKY kappa hyperparameters, which uses the same data to fix a prior. No invented entities are introduced.

free parameters (6)
  • HKY transition/transversion base distribution mean = posterior mean of kappa from HKY+No Variation analysis
    Set via empirical Bayes to fix the base distribution for kappa in HKY infinite mixtures (Section 3).
  • HKY transition/transversion base distribution SD multiplier = 10x posterior SD of kappa from HKY+No Variation analysis
    Chosen to keep the base distribution vague while numerically stable (Section 3).
  • Number of gamma rate categories in standard Yang 1994 model = 5
    Fixed number of equally probable rate categories for Gamma and Codon+Gamma models (Section 3).
  • Data squashing group size D = 1 to about 10% of alignment sites
    Algorithm tuning parameter chosen by hand for joint site-assignment updates (Section 2.4).
  • HCV root height prior = normal(mean=1.0, sd=0.1)
    Restrictive prior needed because substitution rate and time are not identifiable when all sampling dates are equal (Section 3.2).
  • Strict clock rate under infinite mixtures = 1.0 (fixed)
    Fixing the clock rate to 1 allows absolute site-specific substitution rates to be inferred; standard models estimate a relative clock (Section 3).
assumptions (11)
  • standard math Stick-breaking representation of the Dirichlet process (Sethuraman, 1994).
    Used in Section 2.1 to define DP(alpha, G0) draws as discrete random measures.
  • standard math Blackwell-MacQueen Polya urn / Chinese restaurant process.
    Used in Section 2.1 to integrate out G and describe site-to-category assignment probabilities.
  • standard math Hierarchical Dirichlet process construction (Teh et al., 2006).
    Used in Sections 2.2 and 2.3 to build HDP and IHMM priors.
  • standard math Infinite hidden Markov model via HDP-HMM equivalence (Beal et al., 2002; Teh et al., 2006).
    Used in Section 2.3 to model spatial correlation along alignments.
  • standard math Felsenstein pruning likelihood for continuous-time Markov chain evolution.
    Assumed in Section 2 to compute P(Xi | tau, theta_i).
  • domain assumption Sites evolve independently conditional on tree and parameters.
    Stated in Section 2: 'We assume that the data X are generated by continuous-time Markov chains that act independently at the different alignment sites.'
  • domain assumption Time-reversible nucleotide substitution models (HKY, GTR) are adequate.
    Used throughout; the paper notes there is no biological reason for time reversibility but adopts it for computational convenience.
  • domain assumption Strict molecular clock with rate fixed to 1.0 under infinite mixtures.
    Section 3: required to model absolute site-specific substitution rates; standard models instead estimate relative rates.
  • domain assumption Codon-position grouping is a natural group structure for the HDP-Codon model.
    Section 3: alignment sites divided into 3 or 4 groups according to coding position.
  • domain assumption Vague gamma priors with shape=rate=0.001 on concentration parameters.
    Section 3: diffuse priors for DP/HDP/IHMM concentration parameters.
  • ad hoc to paper Empirical Bayes prior for HKY kappa base distribution.
    Section 3: the base distribution hyperparameters are set to estimated mean and 10 times estimated SD from HKY+No Variation analyses; this is data-dependent and specific to this paper.

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Pith. "Pith review of Infinite Mixture Models for Improved Modeling of Across-Site Evolutionary Variation." pith.science (2026). https://pith.science/paper/J5AELACZ

@misc{pith2026241206042,
  author       = {Pith},
  title        = {Pith review of: Infinite Mixture Models for Improved Modeling of Across-Site Evolutionary Variation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J5AELACZ}},
  note         = {Machine review of arXiv:2412.06042}
}
abstract

Scientific studies in many areas of biology routinely employ evolutionary analyses based on the probabilistic inference of phylogenetic trees from molecular sequence data. Evolutionary processes that act at the molecular level are highly variable, and properly accounting for heterogeneity in evolutionary processes is crucial for more accurate phylogenetic inference. Nucleotide substitution rates and patterns are known to vary among sites in multiple sequence alignments, and such variation can be modeled by partitioning alignments into categories corresponding to different substitution models. Determining $\textit{a priori}$ appropriate partitions can be difficult, however, and better model fit can be achieved through flexible Bayesian infinite mixture models that simultaneously infer the number of partitions, the partition that each site belongs to, and the evolutionary parameters corresponding to each partition. Here, we consider several different types of infinite mixture models, including classic Dirichlet process mixtures, as well as novel approaches for modeling across-site evolutionary variation: hierarchical models for data with a natural group structure, and infinite hidden Markov models that account for spatial patterns in alignments. In analyses of several viral data sets, we find that different types of infinite mixture models emerge as the best choices in different scenarios. To enable these models to scale efficiently to large data sets, we adapt efficient Markov chain Monte Carlo algorithms and exploit opportunities for parallel computing. We implement this infinite mixture modeling framework in BEAST X, a widely-used software package for Bayesian phylogenetic inference.

Figures

Figures reproduced from arXiv: 2412.06042 by the authors.

Figure 4
Figure 4. There is generally substantial variation, but the patterns under the GTR [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 1
Figure 1. Performance of different models for across-site variation on respiratory [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
Figure 2
Figure 2. Summary of alignment site clustering patterns inferred from infinite mix [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figures from the paper (2 more)
Figure 3
Figure 3. Figure 3: Summary of alignment site clustering patterns inferred from infinite mix [PITH_FULL_IMAGE:figures/full_fig_p023_3.png]
Figure 4
Figure 4. Figure 4: Summary of alignment site clustering patterns inferred from infinite mixture [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]

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Works this paper leans on

107 extracted references · 79 canonical work pages

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address author booktitle chapter edition editor eid howpublished institution journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := ...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize ":" * " " *...

  3. [3]

    Akaike H. 1973. Information theory and an extension of the maximum likelihood principle. In: Petrov BN, Csaki F, editors, Proceedings of the Second International Symposium on Information Theory. pp. 267--281

    1973

  4. [4]

    Aldous D. 1985. Ecole d'Ete de Probabilities de Saint-Flour XIII-1983, Springer-Verlag, chapter Exchangeability and related topics, pp. 1--198

    1985

  5. [5]

    Antoniak C. 1974. Mixtures of D irichlet processes with applications to B ayesian nonparametric problems. The Annals of Statistics. 2:1152--1174

    1974

  6. [6]

    Attwood SW, Hill SC, Aanensen DM, Connor TR, Pybus OG. 2022. Phylogenetic and phylodynamic approaches to understanding annd combating the early SARS-CoV-2 pandemic. Nature Reviews Genetics. 23:547--562

    2022

  7. [7]

    Ayres DL, Cummings MP, Baele G, Darling AE, Lewis PO, Swofford DL, Huelsenbeck JP, Lemey P, Rambaut A, Suchard MA. 2019. BEAGLE 3: Improved performance, scaling, and usability for a high-performance computing library for statistical phylogenetics. Systematic Biology. 68:1052--1061

    2019

  8. [8]

    Baele G, Gill MS, Bastide P, Lemey P, Suchard MA. 2021. Markov-modulated continuous-time markov chains to identify site- and branch-specific evolutionary variation in BEAST . Systematic Biology. 70:181--189

    2021

Show all 107 references
  1. [9]

    Baele G, Lemey P, Bedford T, Rambaut A, Suchard MA, Alekseyenko AV. 2012 a . Improving the accuracy of demographic and molecular clock model comparison while accommodating phylogenetic uncertainty. Molecular Biology and Evolution. 29:2157--67

    2012

  2. [10]

    Baele G, Lemey P, Suchard MA. 2016. Genealogical working distributions for B ayesian model testing with phylogenetic uncertainty. Systematic Biology. 65:250--264

    2016

  3. [11]

    Baele G, Li WLS, Drummond AJ, Suchard MA, Lemey P. 2012 b . Accurate model selection of relaxed molecular clocks in B ayesian phylogenetics. Molecular Biology and Evolution. 20:239--243

    2012

  4. [12]

    Basu S, Chib S. 2003. Marginal likelihood and B ayes factors for D irichlet process mixture models. Journal of the American Statistical Association. 98:224--235

    2003

  5. [13]

    Baum LE, Petrie T. 1966. Statistical inference for probabilistic functions of finite state M arkov chains. Annals of Mathematical Statistics. 37:1554--1563

    1966

  6. [14]

    Beal MJ, Ghahramani Z, Rasmussen C. 2002. The infinite hidden M arkov model. In: Dietterich TG, Becker S, Ghahramani Z, editors, Advances in Neural Information Processing Systems. MIT Press, volume 14, pp. 577--584

    2002

  7. [15]

    Biek R, Henderson J, Waller L, Rupprecht C, Real L. 2007. A high-resolution genetic signature of demographic and spatial expansion in epizootic rabies virus. Proceedings of the National Academy of Sciences. 104:7993--7998

    2007

  8. [16]

    Blackwell D, MacQueen J. 1973. Ferguson distributions via polya urn schemes. The Annals of Statistics. 1:353--355

    1973

  9. [17]

    Bletsa M, Suchard MA, Ji X, Gryseels S, Vrancken B, Baele G, Worobey M, Lemey P. 2019. Divergence dating using mixed effects clock modelling: an application to HIV-1 . Virus Evolution. 5:vez036

    2019

  10. [18]

    Bruno WJ. 1996. Modeling residue usage in aligned protein sequences via maximum likelihood. Molecular Biology and Evolution. 13:1368--1374

    1996

  11. [19]

    Crotty SM, Holland BR. 2022. Comparing partitioned models to mixture models: do information criteria apply? Systematic Biology. 71:1541--1548

    2022

  12. [20]

    Didelot X, Siveroni I, Volz EM. 2021. Additive uncorrelated relaxed clock models for the dating of genomic epidemiology phylogenies. Molecular Biology and Evolution. 38:307--317

    2021

  13. [21]

    Douglas J, Geoghegan JL, Hadfield J, Bouckaert R, Storey M, Ren X, de Ligt J, French N, Welch D. 2021. Real-time genomics for tracking severe acute respiratory syndrome coronavirus 2 border incursions after virus elimination, N ew Z ealand. Emerging Infectious Diseases. 27:2361--2368

    2021

  14. [22]

    Drummond A, Ho S, Phillips M, Rambaut A. 2006. Relaxed phylogenetics and dating with confidence. PLoS Biology. 4:e88

    2006

  15. [23]

    Drummond AJ, Suchard MA. 2010. Bayesian random local clocks, or one rate to rule them all. BMC Biology. 8

    2010

  16. [24]

    Fan Y, Wu R, Chen MH, Kuo L, Lewis PO. 2011. Choosing among partition models in B ayesian phylogenetics. Molecular Biology and Evolution. 28:523--532

    2011

  17. [25]

    Felsenstein J. 1981. Evolutionary trees from DNA sequences: a maximum likelihood approach. Journal of Molecular Evolution. 13:93--104

    1981

  18. [26]

    Felsenstein J. 2004. Inferring Phylogenies. Sunderland, MA: Sinauer Associates, Inc

    2004

  19. [27]

    Felsenstein J, Churchill G. 1996. A hidden M arkov model approach to variation among sites in rate of evolution. Molecular Biology and Evolution. 13:93--104

    1996

  20. [28]

    Ferguson T. 1973. A B ayesian analysis of some nonparametric problems. The Annals of Statistics. 1:209--230

    1973

  21. [29]

    Fourment M, Claywell BC, Dinh V, McCoy C, Matsen IV FA, Darling AE. 2018. Effective online B ayesian phylogenetics via sequential M onte C arlo with guided proposals. Systematic Biology. 67:490--502

    2018

  22. [30]

    Fourment M, Magee AF, Whidden C, Bilge A, Matsen IV FA, Minin VN. 2020. 19 dubious ways to compute the marginal likelihood of a phylogenetic tree topology. Systematic Biology. 69:209--220

    2020

  23. [31]

    Fox EB, Sudderth EB, Jordan MI, Willsky AS. 2011. A sticky HDP-HMM with application to speaker diarization. The Annals of Applied Statistics. 5:1020--1056

    2011

  24. [32]

    Galtier N. 2001. Maximum-likelihood phylogenetic analysis under a covarion-like model. Molecular Biology and Evolution. 18:866--873

    2001

  25. [33]

    Gascuel O, Guindon S. 2007. Reconstructing evolution: new mathematical and computational advances, Oxford University Press, chapter Modelling the variability of evolutionary processes

    2007

  26. [34]

    Gelfand AE, Kottas A, MacEachern SN. 2005. Bayesian nonparametric spatial modeling with D irichlet process mixing. Journal of the American Statistical Association. 100:1021--1035

    2005

  27. [35]

    Gelman A, Hwang J, Vehtari A. 2014. Understanding predictive information criteria for B ayesian models. Statistics and Computing. 24:997--1016

    2014

  28. [36]

    Gelman A, Meng XL. 1998. Simulating normalizing constants: from importance sampling to bridge sampling to path sampling. Statistical Science. 13:163--185

    1998

  29. [37]

    Ghahramani Z. 2013. Bayesian non-parametrics and the probabilistic approach to modelling. Philosophical Transactions of the Royal Society A. 371:20110553

    2013

  30. [38]

    Gill MS, Lemey P, Faria NR, Rambaut A, Shapiro B, Suchard MA. 2013. Improving B ayesian population dynamics inference: a coalescent-based model for multiple loci. Molecular Biology and Evolution. 30:713--724

    2013

  31. [39]

    Gill MS, Lemey P, Suchard MA, Rambaut A, Baele G. 2020. Online B ayesian phylodynamic inference in BEAST with application to epidemic reconstruction. Molecular Biology and Evolution. 37:1832--1842

    2020

  32. [40]

    Golding GB. 1983. Estimates of DNA and protein sequence divergence: an examination of some assumptions. Molecular Biology and Evolution. 1:125--142

    1983

  33. [41]

    Guha S. 2010. Posterior simulation in countable mixture models for large datasets. Journal of the American Statistical Association. 105:775--786

    2010

  34. [42]

    Guindon S, Rodrigo AG, Dyer KA, Huelsenbeck JP. 2004. Modeling the site-specific variation of selection patterns along lineages. Proceedings of the National Academy of Sciences. 101:12957--12962

    2004

  35. [43]

    Hairault A, Robert CP, Rousseau J. 2022. Evidence estimation in finite and infinite mixture models and applications. ArXiv:2205.05416

    2022 arXiv

  36. [44]

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

    1985

  37. [45]

    Hastings W. 1970. M onte C arlo sampling methods using M arkov chains and their applications. Biometrika. 57:97--109

    1970

  38. [46]

    Heath TA, Holder MT, Huelsenbeck JP. 2012. A D irichlet process prior for estimating lineage-specific substitution rates. Molecular Biology and Evolution. 29:939--955

    2012

  39. [47]

    Hillis DM, Heath TA, St John K. 2005. Analysis and visualization of tree space. Systematic Biology. 54:471--482

    2005

  40. [48]

    Huelsenbeck JP. 2002. Testing a covariotide model of DNA substitution. Molecular Biology and Evolution. 19:698--707

    2002

  41. [49]

    Huelsenbeck JP, Larget B, Swofford DL. 2000. A compound P oisson orocess for relaxing the molecular clock. Genetics. 154:1879--1892

    2000

  42. [50]

    Huelsenbeck JP, Nielsen R. 1999. Variation in the pattern of nucleotide substitution across sites. Journal of Molecular Evolution. 48:86--93

    1999

  43. [51]

    Huelsenbeck JP, Suchard MA. 2007. A nonparametric method for accommodating and testing across-site rate variation. Systematic Biology. 56:975--987

    2007

  44. [52]

    Jefferys W, Berger J. 1992. Ockham's razor and B ayesian analysis. American Statistician. 80:64--72

    1992

  45. [53]

    Jeffreys H. 1935. Some tests of significance, treated by the theory of probability. Mathematical Proceedings of the Cambridge Philosophical Society. 31:203--222

    1935

  46. [54]

    Jeffreys H. 1961. Theory of Probability. Oxford University Press

    1961

  47. [55]

    Jukes T, Cantor C. 1969. Evolution of protein molecules. In: Munro H, editor, Mammalian Protein Metabolism. Academic Press, pp. 21--132

    1969

  48. [56]

    Kass R, Raftery A. 1995. B ayes factors. Journal of the American Statistical Association. 90:773--795

    1995

  49. [57]

    Kimura M. 1968. Evolutionary rate at the molecular level. Nature. 217:624--626

    1968

  50. [58]

    Kimura M. 1980. A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences. Journal of Molecular Evolution. 16:111--120

    1980

  51. [59]

    Lakner C, van der Mark P, Huelsenbeck B J P annd Larget, Ronquist F. 2008. Efficiency of M arkov chain M onte C arlo tree proposals in B ayesian phylogenetics. Systematic Biology. 57:86--103

    2008

  52. [60]

    Lanave C, Preparata G, Saccone C, Serio G. 1984. A new method for calculating evolutionary substitution rates. Jounal of Molecular Evolution. 20:86--93

    1984

  53. [61]

    Lartillot N, Philippe H. 2004. A B ayesian mixture model for across-site heterogeneities in the amino-acid replacement process. Molecular Biology and Evolution. 21:2004

    2004

  54. [62]

    Lartillot N, Philippe H. 2006. Computing B ayes factors using thermodynamic integration. Systematic Biology. 55:195--207

    2006

  55. [63]

    Lepage T, Lawi S, Tupper P, Bryant D. 2006. Continuous and tractable models for the variation of evolutionary rates. Mathematical Biosciences. 199:216--233

    2006

  56. [64]

    Lindsey M, Weare J, Zhang A. 2022. Ensemble M arkov chain M onte C arlo with teleporting walkers. SIAM/ASA Journal on Uncertainty Quantification. 10:860--885

    2022

  57. [65]

    Liu Q, Charleston MA, Richards SA, Holland BR. 2023. Performance of A kaike information criterion and B ayesian information criterion in selecting partition models and mixture models. Systematic Biology. 72:92--105

    2023

  58. [66]

    MacEachern SN. 1999. Dependent nonparametric processes. In: ASA P roceedings of the S ection on B ayesian S tatistical S cience. pp. 50--55

    1999

  59. [67]

    MacEachern SN. 2000. Dependent D irichlet processes. Technical report, Ohio State University, Deparment of Statistics

    2000

  60. [68]

    Magee AF, Holbrook AJ, Pekar JE, Caviedes-Solis IW, Matsen Iv FA, Baele G, Wertheim JO, Ji X, Lemey P, Suchard MA. 2024. Random-effects substitution models for phylogenetics via scalable gradient approximations. Syst Biol. 73:562--578

    2024

  61. [69]

    Metropolis N, Rosenbluth A, Rosenbluth M, Teller A, Teller E. 1953. Equation of state calculation by fast computing machines. Journal of Chemical Physics. 21:1087--1092

    1953

  62. [70]

    Neal RM. 2000. M arkov chain sampling methods for D irichlet process mixture models. Journal of Computational and Graphical Statistics. 9:249--265

    2000

  63. [71]

    Nei M, Chakraborty R, Fuerst PA. 1976. Infinite allele model with varying mutation rate. Proceedings of the National Academy of Sciences. 73:4164--4168

    1976

  64. [72]

    Newton M, Raftery A. 1994. Approximate B ayesian inference with the weighted likelihood bootstrap. Journal of the Royal Statistical Society, Series B. 56:3--48

    1994

  65. [73]

    Nielsen R. 1997. Site-by-site estimation of the rate of evolution and the correlation of rates in mitochondrial DNA . Systematic Biology. 46:346--353

    1997

  66. [74]

    Pagel M, Meade A. 2004. A phylogenetic mixture model for detecting pattern-heterogeneity in gene sequence or character-state data. Systematic Biology. 53:571--581

    2004

  67. [75]

    Paisley J, Wang C, Blei DM, Jordan MI. 2014. Nested hierarchical D irichlet processes. IEEE transactions on pattern analysis and machine intelligence. 37:256--270

    2014

  68. [76]

    Pitman J, Yor M. 1997. The two-parameter P oisson- D irichlet distribution derived from a stable subordinator. The Annals of Probability. 25:855--900

    1997

  69. [77]

    (101 co-authors)

    Quick J, Loman N, Durrafour S, et al. (101 co-authors). 2016. Real-time, portable genome sequencing for E bola surveillance. Nature. 530:228--232

    2016

  70. [78]

    R Core Team . 2021. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria

    2021

  71. [79]

    Raftery A, Newton M, Satagopan J, Krivitsky P. 2007. Bayesian Statistics, Oxford University Press, chapter Estimating the integrated likelihood via posterior simulation using the harmonic mean identity, pp. 1--45

    2007

  72. [80]

    Ray SC, Arthur RR, Carella A, Bukh J, Thomas DL. 2000. Genetic epidemiology of hepatitis C virus throughout E gypt. Journal of Infectious Diseases. 182:698--707

    2000

  73. [81]

    Redelings BD, Suchard MA. 2005. Joint B ayesian estimation of alignment and phylogeny. Systematic Biology. 54:401--418

    2005

  74. [82]

    Rodriguez A, Dunson DB, Gelfand AE. 2008. The nested D irichlet process. Journal of the American Statistical Association. 103:1131--11144

    2008

  75. [83]

    Sethuraman J. 1994. A constructive definition of D irichlet priors. Statistica Sinica. 4:639--650

    1994

  76. [84]

    Suchard MA, Lemey P, Baele G, Ayres DL, Drummond AJ, Rambaut A. 2018. Bayesian phylogenetic and phylodynamic data integration using BEAST 1.10. Virus Evolution. 4:vey016

    2018

  77. [85]

    Susko E, Roger AJ. 2020. On the use of information criteria for model selection in phylogenetics. Molecular Biology and Evolution. 37:549--562

    2020

  78. [86]

    Swofford DL, Olsen GJ, Waddell PJ, Hillis DM. 1996. Molecular Systematics, Sinauer Associates, Inc., chapter Phylogenetic Inference, pp. 407--514. 2nd edition

    1996

  79. [87]

    Tamura K. 1992. Estimation of the number of nucleotide substitutions when there are stronng transition-transversion and G + C -content biases. Molecular Biology and Evolution. 9:678--687

    1992

  80. [88]

    Tamura K, Nei M. 1993. Estimation of the number of nucelotide substitutions in the control region of mitochondrial DNA in humans and chimpanzees. Molecular Biology and Evolution. 10:512--526

    1993

  81. [89]

    Tavare S. 1986. Some probabilistic and statistical problems on the analysis of dna sequences. Lectures on Mathematics in the Life Sciences. 17:57--86

    1986

  82. [90]

    Teh YW, Jordan MI, Beal MJ, Blei DM. 2006. Hierarchical D irichlet processes. Journal of the American Statistical Association. 101:1566--1581

    2006

  83. [91]

    Thorne J, Kishino H, Painter I. 1998. Estimating the rate of evolution of the rate of molecular evolution. Molecular Biology and Evolution. 15:1647--1657

    1998

  84. [92]

    Tuffley C, Steel M. 1998. Modeling the covarion hypothesis of nucleotide substitution. Mathematical Biosciences. 147:63--91

    1998

  85. [93]

    Venables WN, Ripley BD. 2002. Modern Applied Statistics with S . New York: Springer, fourth edition

    2002

  86. [94]

    Venditti C, Meade A, Pagel M. 2008. Phylogenetic mixture models can reduce node-density artifacts. Systematic Biology. 57:286--293

    2008

  87. [95]

    Waddell PJ, Steel MA. 1997. General time-reversible distances with unequal rates across sites: mixing and inverse G aussian distributions with invariant sites. Molecular Phylogenetics and Evolution. 8:398--414

    1997

  88. [96]

    Warren DL, Geneva AJ, Lanfear R. 2017. RWTY: (R We There Yet): an R package for examining convergence of B ayesian phylogenetic analyses. Molecular Biology and Evolution. 34:1016--1020

    2017

  89. [97]

    Watanabe S. 2010. Asymptotic equivalence of B ayess cross validation and widely applicable information criterion in singular learning theory. Journal of Machine Learning Research. 11:3571--3594

    2010

  90. [98]

    Whelan S. 2008. Spatial and temporal heterogeneity in nuceleotide sequence evolution. Molecular Biology and Evolution. 25:1683--1694

    2008

  91. [99]

    Wu CH, Suchard MA, Drummond AJ. 2013. B ayesian selection of nucleotide substitution model and their site assignments. Molecular Biology and Evolution. 30:669--688

    2013

  92. [100]

    Xie W, Lewis PO, Kuo MH L annd Chen. 2011. Improving marginal likelihood estimation for B ayesian phylogenetic model selection. Systematic Biology. 60:150--160

    2011

  93. [101]

    Yang GL, Le Cam L. 2000. Asymptotics in Statistics: Some Basic Concepts. Berlin: Springer

    2000

  94. [102]

    Yang Z. 1993. Maximum-likelihood estimation of phylogeny from DNA sequences when substitution rates differ over sites. Molecular Biology and Evolution. 10:1396--1401

    1993

  95. [103]

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

    1994

  96. [104]

    Yang Z. 1995. A space-time process model for the evolution of DNA sequences. Genetics. 139:993--1005

    1995

  97. [105]

    Yang Z. 2006. Computational Molecular Evolution. Oxford University Press

    2006

  98. [106]

    Zhou Y, Brinkmannn H, Rodrigue N, Lartillot N, Philippe H. 2010. A D irichlet process covarion mixture model and its assessments using posterior predictive discrepancy tests. Molecular Biology and Evolution. 27:371--384

    2010

  99. [107]

    Zlateva K, Lemey P, Mo \"e s E, Vandamme AM, Van Ranst M. 2005. Genetic variability and molecular evolution of the human respiratory syncytial virus subgroup B atttachment G protein. J. Virol. 79:9157--9167

    2005

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.