Pith. sign in

REVIEW 3 major objections 7 minor 127 references

Mixing efficiency of trans-model Markov chain Monte Carlo algorithms with applications in Bayesian phylogenetics

T0 review · 3 major / 7 minor · reviewed 2026-07-09 · glm-5.2

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Entrance distribution governs trans-model MCMC mixing efficiency

desk verdict Letter to colleague on trans-model MCMC mixing paper read the letter →

arxiv 2607.07188 v1 pith:4RUIRKVN submitted 2026-07-08 stat.CO

Xiyun Jiao , Thomas Flouris , Ziheng Yang This is my paper
classification stat.CO
keywords algorithmsmcmctrans-modelbayesianefficiencymixinganalysiscarlo
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 studies when and why trans-model MCMC algorithms—the kind that jump between distinct statistical models, such as different phylogenetic trees—mix well or poorly. The central object is the entrance distribution: the distribution of parameter values immediately after the chain arrives in a new model. The authors prove that for two-model comparisons, maximum mixing efficiency is achieved if and only if this entrance distribution matches the within-model posterior of the new model. The analysis reveals an asymmetry. When the simpler model has posterior probability below one-half, achieving the maximum jump rate is necessary but not sufficient for optimal mixing: the proposal density must either already equal the posterior, or a within-model move must be added to restore the parameter to its stationary distribution. When the simpler model has posterior probability at or above one-half, the acceptance step itself thins the proposal to match the target, so maximum jump rate becomes both necessary and sufficient, and a range of proposal densities can achieve optimal efficiency without any within-model move. The paper also shows that for more than two models, the optimal chain does not maximize the jump rate—proposing too many model switches can hurt efficiency. These principles are tested on two real phylogenetic datasets, where direct transfer of branch lengths between trees and preferential perturbation of short internal branches substantially improve cross-tree mixing.

What carries the argument

The analysis proceeds through two exactly solvable examples (uniform and Gaussian target distributions) for the two-model case, where one model has no free parameters. The entrance distribution is defined and shown to be determined by the proposal density and the acceptance ratio. For the general two-model case, the parameter space of the parametric model is discretized into bins, reducing the problem to a finite-state Markov chain whose optimal transition matrix is known from prior work. Taking the discretization to the continuum yields the optimal transition kernel and the conditions on the proposal density. For more than two models, the theory of super-efficient reversible chains (via the

What would settle it

If, in a two-model setting where the dominant model has posterior probability below one-half, one could exhibit a proposal density g(·) ≠ π₁(·) that achieves both maximum jump rate and maximum efficiency without any within-model move, the claimed necessity of the entrance-distribution matching condition would be refuted.

Watch

Extended reading notes

Core claim

The entrance distribution—the distribution of parameters immediately upon entering a new model—must match the within-model posterior for a trans-model MCMC algorithm to achieve maximum mixing efficiency. This single condition explains why some algorithms with high acceptance rates still mix poorly, and why adding within-model parameter updates sometimes helps and sometimes does not. The condition also implies that maximum model-jump probability is sufficient for optimal efficiency only when the dominant model has posterior probability at or above one-half; otherwise it is necessary but not sufficient.

Load-bearing premise

The theoretical results are derived for the case of exactly two models where one has no free parameters; the extension to the many-model, many-parameter setting of phylogenetics relies on an unproven analogy.

Editorial extensions

If this is right

  • Algorithm designers should prioritize matching the entrance distribution to the within-model posterior when constructing cross-model proposals, rather than simply maximizing the acceptance rate.
  • In phylogenetic MCMC, direct transfer of branch lengths between trees is preferable to merge-and-split proposals because it produces values closer to the posterior mode, confirming the entrance-distribution principle in practice.
  • The finding that maximum jump rate is not always optimal for more than two models suggests that aggressive model-switching proposals in software with many candidate models may be counterproductive.
  • The deterioration of mixing as dataset size increases—because fixed proposals drift from an increasingly concentrated posterior—points to a need for adaptive proposals that track the posterior as data accumulate.
  • The distinction between within-model and cross-model moves implies that overall acceptance rate is a misleading performance metric for mixed algorithms; cross-model and within-model acceptance rates should be reported separately.

Reading between the lines

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

  • If the entrance-distribution principle extends to cases where both models have free parameters, then cross-model proposals should propose all parameters of the new model jointly from an approximation to their conditional posterior, not independently or from the prior.
  • The result that the acceptance step can thin a non-matching proposal to match the target (when the dominant model has probability above one-half) suggests a connection to rejection sampling that could be exploited to design proposals with lower computational cost per accepted move.
  • For phylogenetic datasets with multiple local peaks in tree space, the entrance-distribution criterion would need to be satisfied within each peak, suggesting that locally informed proposals conditioned on the current tree neighborhood may be more effective than global proposals.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 7 minor

Summary. This paper studies the mixing efficiency of trans-model MCMC algorithms, with particular attention to Bayesian phylogenetics where tree topologies constitute different models. The authors analyze two simple but illuminating examples (uniform and Gaussian target distributions) involving two models, one of which (H0) has no free parameters. They derive conditions under which the model-jump probability P_jump and the mixing efficiency E reach their maxima, and characterize when within-model moves are necessary to achieve optimal efficiency. The key theoretical results state that: (i) P_jump is bounded above by 2(1−max{π_k}) (Theorem 1); (ii) when π_0 < 1/2, maximum P_jump is necessary but not sufficient for maximum E, and within-model moves restoring parameters to their stationary distribution are needed unless the proposal g(·) already equals the posterior π_1(·); (iii) when π_0 ≥ 1/2, maximum P_jump is both necessary and sufficient for maximum E. The K > 2 discrete case is treated via the framework of Frigessi et al. (1992). The theory is then applied to NNI and SPR tree-proposal algorithms using two real datasets (primate ψη-globin pseudogenes and mammalian mitochondrial genes), testing strategies for branch-length proposals and tree-proposal weights.

Significance. The paper addresses a genuinely under-studied problem: the mixing efficiency of cross-model MCMC moves, as opposed to the well-studied within-model case. The mathematical analysis for the two-model case is clean and self-contained, with explicit derivations for both uniform and Gaussian examples (SI texts 2–3) and a correct proof of Theorem 1 (SI text 1). The discrete-state optimality results (SI text 4) correctly build on Frigessi et al. (1992). The practical guidelines—preferentially propose high-posterior models, propose parameters near the posterior mode, use direct transfer of branch lengths rather than merge-and-split—are sensible and supported by the empirical results (Tables 2–3), where NNI with direct transfer outperforms SPR baseline by factors of 16–21. The paper provides falsifiable, quantitative predictions. The distinction drawn between within-model and trans-model MCMC (e.g., that intermediate acceptance rates are optimal for within-model moves but high acceptance rates are desirable for cross-model moves) is a useful conceptual contribution.

major comments (3)
  1. The central theoretical results (eqs. 28–38) are derived for exactly two models where H0 has no free parameters. The extension to K > 2 models (SI text 4) explicitly assumes parameters are proposed from the posterior during cross-model moves (q(θ_{k'}|k,θ_k,k') = π(θ_{k'}|k')), which is the ideal the paper aims to justify. The phylogenetic applications involve K >> 2 trees, each with 2n−3 branch-length parameters, where proposing from the posterior is infeasible. The authors acknowledge this gap (Discussion: 'General cases where both models have free parameters, or where more than two models with parameters are under comparison, are yet to be studied'), but the abstract and discussion present the phylogenetic guidelines as flowing from the theory. The paper would benefit from explicitly stating, early in the phylogenetics section, that the connection between the two-model theory and theK
  2. The empirical results in Tables 2–3 are based on only two datasets. The ψη-globin dataset has 6 species (105 possible trees) with the top three trees accounting for ~100% of the posterior, while the mt dataset has 29 species with the top two trees having posteriors 0.574 and 0.107. These are quite different scenarios, yet the relative performance of algorithms is broadly similar across both. It would strengthen the paper to discuss whether the theoretical conditions (e.g., π_0 ≥ 1/2 vs. π_0 < 1/2) are met in each dataset, and whether the empirical findings are consistent with the theory's predictions for each case. Currently, the link between the theoretical conditions and the empirical settings is not made explicit.
  3. The branch weight exponent in eq. S90 (w_f ∝ t_f^{-1/2}) and the parsimony weight scale parameter a in eq. S91 (w_{k'} ∝ exp((S_k − S_{k'})/a), with a = 0.5 chosen after testing values 0.1–20) are free parameters tuned on the same datasets used for evaluation. The paper does not discuss whether this tuning constitutes overfitting or how the optimal values might transfer to other datasets. The authors note that 'our branch weight (eq. S90) is arbitrary and may suit one dataset better than another' (Results section), but the parsimony weight tuning receives no such caveat. A brief discussion of the sensitivity to these choices would be appropriate.
minor comments (7)
  1. The notation π_0 is used for the posterior probability of H0, but π_1(θ) is used for the posterior density of θ within H1. This is consistent but occasionally confusing; a clarifying note on first use would help.
  2. In the caption of Figure 4, the note states 'the round flat tops in (a) and (c) are the same, the apparent differences being due to numerical issues in calculation of E and in kernel density smoothing.' If numerical issues affect the figure, consider whether the figure can be improved or whether the statement should be more prominent.
  3. The reference to 'Liet al., 2000' in the Introduction appears to be 'Li et al., 2000' with a missing space. Several other citation formatting issues exist (e.g., 'Chenet al., 2014', 'Ronquistet al., 2012').
  4. In the Discussion, the statement 'it is common to run an MCMC algorithm for N = 10^9 iterations, say, achieving an ESS of <100, in which case E < 10^{-7}' is presented without citation. A reference or caveat would strengthen this claim.
  5. Figure 8 caption: 'The cross-tree move is B4 (NNI bw) while the within-tree move changes one randomly sampled branch length.' It would help to state the y-axis units more explicitly in the caption (e.g., 'Time(s)/100' for panel a and 'Time(h) 1000' for panel b are somewhat cryptic).
  6. The paper mentions Lindley's paradox in the context of Figure 5 but does not elaborate on its relevance to the mixing efficiency problem. A sentence connecting the paradox to the practical implications for trans-model MCMC would improve clarity.
  7. SI text 2 derives autocorrelation functions for the case π_0 < 1/2 with maximum P_jump but suboptimal E. The derivation is detailed but the key insight—that ρ_1 achieves its optimal value but ρ_k for k ≥ 2 do not—could be stated more prominently at the beginning of the section rather than only in the surrounding main text.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: derivations are self-contained from standard MCMC theory; one minor self-referential assumption in K>2 extension

  1. self definitional [SI text 4, eqs. 28-29 and surrounding text]
    "the theory also applies when the parameters for the new model are proposed from the posterior during the cross-model move, that is, if q(θ_{k'}|k, θ_k, k') = π(θ_{k'}|k') ≡ p(θ_{k'}|k', X)."

    The K>2 optimality results (eqs. 28-29) are derived under the assumption that parameters are proposed from the posterior during cross-model moves. This is the very ideal the paper advocates ('propose parameter values from the posterior as much as possible'). However, the paper is transparent about this: it presents this as a known special case, not as evidence for the general claim. The two-model results (the paper's main theoretical contribution) are derived without this assumption. The K>2 extension is labeled as applying 'when' the ideal condition holds, not as proving that the ideal is optimal. This is a limitation of scope, not a circular derivation.

full rationale

The paper's core theoretical results (eqs. 12-38) are derived self-containedly from the Metropolis-Hastings acceptance ratio (eq. 2) and standard Markov chain spectral theory (SI text 4, citing Frigessi et al. 1992 and Peskun 1973). The optimality conditions for two-model trans-model MCMC emerge from analyzing the transition kernel and autocorrelation structure, not from assuming the conclusion. The K>2 extension (SI text 4) does assume posterior-proposal of parameters, which is the ideal the paper advocates, but the paper explicitly frames this as a special case ('the theory also applies when...') rather than using it to prove the ideal's optimality. The two-model results, which constitute the paper's main theoretical contribution, do not rely on this assumption. The phylogenetic applications are presented as empirical illustrations guided by the theory's principles, with the paper acknowledging the gap between the two-model theory and the K>>2 phylogenetic setting. No prediction reduces to a fitted input by construction, and no central claim depends on a self-citation chain. The one identified step is a scope limitation acknowledged by the authors, not a circular derivation.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new mathematical entities or physical constructs. The free parameters are tuning constants in the phylogenetic proposal algorithms, not in the theoretical derivations. The theoretical results are parameter-free (they depend only on π₀ and the proposal density g). The axioms are standard MCMC theory plus one heuristic bridge assumption connecting the two-model theory to multi-model phylogenetic practice.

free parameters (3)
  • Branch weight exponent (eq. S90) = -1/2
    The weight w_f ∝ t_f^{-1/2} for sampling internal branches is stated as arbitrary (Discussion: 'our branch weight is arbitrary and may suit one dataset better than another'). It is not derived from theory.
  • Parsimony weight scale parameter a (eq. S91) = 0.5
    The parameter a in w_{k'} ∝ exp((S_k - S_{k'})/a) was tested at values 0.1, 0.5, 1, 5, 10, 20 and a=0.5 was selected for good performance. This is a tuning parameter chosen empirically.
  • NNI focal branch modification window size (B2, B3) = 0.005
    The window size ε=0.005 for multiplier moves on focal branch lengths in NNI variants B2 and B3 is set by hand.
assumptions (4)
  • standard math The Metropolis-Hastings acceptance probability min{1, A(ω,ω')} with the Hastings ratio of eq. 2 produces a π-reversible Markov chain with stationary distribution π(k, θ_k).
    Standard MCMC theory (Metropolis et al. 1953; Hastings 1970), invoked throughout the paper as the foundation for all algorithms.
  • standard math The asymptotic variance of the MCMC estimator is ν = ν_f[1 + 2(ρ₁ + ρ₂ + ...)] and efficiency is E = ν_f/ν (eqs. 10-11).
    Standard Markov chain theory (Peskun 1973), used as the definition of mixing efficiency throughout.
  • domain assumption The optimal π-reversible Markov chain for estimating π₁ in the discrete case is characterized by Frigessi et al. (1992), Proposition 1.
    The K > 2 model results (eqs. 28-29, SI text 4) are built on this result. The paper provides a simplified re-derivation but the core optimality characterization is external.
  • ad hoc to paper The two-model theory (one model parameter-free) extends heuristically to phylogenetic tree search with K >> 2 models and many parameters per model.
    The phylogenetic applications involve 105 or astronomically many trees, each with 2n-3 branch lengths. The theory covers only K=2 with one parameter-free model. The paper states this gap explicitly but applies the principles anyway.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Mixing efficiency of trans-model Markov chain Monte Carlo algorithms with applications in Bayesian phylogenetics." pith.science (2026). https://pith.science/paper/4RUIRKVN

@misc{pith2026260707188,
  author       = {Pith},
  title        = {Pith review of: Mixing efficiency of trans-model Markov chain Monte Carlo algorithms with applications in Bayesian phylogenetics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4RUIRKVN}},
  note         = {Machine review of arXiv:2607.07188}
}
read the original abstract

Trans-model Markov chain Monte Carlo (MCMC) algorithms are widely used in Bayesian inference, and are particularly important in Bayesian phylogenetics where phylogenetic trees represent different statistical models. While the algorithm allows great flexibility, its mixing efficiency can vary hugely, and is poorly understood. Here we use mathematical analysis and simulation to explore the mixing efficiency of trans-model MCMC proposals, including the model-proposal probabilities and the proposal kernel for model parameters. Our analysis confirms the intuition that one should preferentially propose models with high posterior probabilities, and propose parameter values from the posterior as much as possible. Our results provide guidelines for constructing efficient trans-model MCMC algorithms. The principles are applied to MCMC algorithms in phylogenetic reconstruction using two real datasets for primates and mammals.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

127 extracted references · 127 canonical work pages

  1. [1]

    , title =

    Lindley, D.V. , title =. Biometrika , volume =

    work page

  2. [2]

    and Bilge, A

    Dinh, V. and Bilge, A. and Zhang, C. and Matsen IV, F.A. , title =. Proceedings of the 34th International Conference on Machine Learning, Sydney, Australia , volume =

    work page

  3. [3]

    and Dunson, D.B

    Nishimura, A. and Dunson, D.B. and Lu, J. , title =. Biometrika , volume =

    work page

  4. [4]

    , title =

    Tierney, L. , title =. Ann. Stat. , volume =

    work page

  5. [5]

    , title =

    Liu, J.S. , title =. Statist. Comput. , volume =

    work page

  6. [6]

    , title =

    Dickey, James M. , title =. Ann. Math. Statist. , volume =. doi:10.1214/aoms/1177693507 , year =

    work page doi:10.1214/aoms/1177693507

  7. [7]

    Miyamoto, M. M. and Slighton, J. L. and Goodman, M. , title =. Science , volume =

    work page

  8. [8]

    and Jimenez-Silva, C

    Douglas, J. and Jimenez-Silva, C. L. and Bouckaert, R. , title =. Syst. Biol. , DOI =

    work page

Show all 127 references
  1. [9]

    , title =

    Tierney, L. , title =. Ann. Stat. , volume =. 1994 , type =

    1994

  2. [10]

    , title =

    Sokal, A.D. , title =. 1989 , type =

    1989

  3. [11]

    Green, P. J. and Han, X. L. , Title =. Stochastic Models, Statistical Methods and Algorithms in Image Analysis , Editor =

  4. [12]

    and Snell, J.L

    Kemeny, J.G. and Snell, J.L. , Title =

  5. [13]

    and Rosenbluth, A

    Metropolis, N. and Rosenbluth, A. W. and Rosenbluth, M. N. and Teller, A. H. and Teller, E. , Journal =. Equation of State Calculations by Fast Computing Machines , Year=

  6. [14]

    and Cantor, C.R

    Jukes, T.H. and Cantor, C.R. , Title =. 1969 , Pages =

    1969

  7. [15]

    , Title =

    Peskun, P.H. , Title =. Biometrika , Volume =

  8. [16]

    Hastings, W. K. , Journal =

  9. [17]

    and Hwang, Chii-Ruey and Younes, Laurent , title =

    Frigessi, Arnoldo. and Hwang, Chii-Ruey and Younes, Laurent , title =. Ann. Appl. Prob. , volume =

  10. [18]

    Geyer, C. J. , Journal =. Practical

  11. [19]

    and Miller, M

    Grenander, U. and Miller, M. I. , Journal =. Representations of knowledge in complex systems , Year=

  12. [20]

    Green, P. J. , Journal =. Reversible jump

  13. [21]

    Carlin, B. P. and Chib, S. , Journal =. Bayesian model choice via

  14. [22]

    and Roberts, G.O

    Gelman, A. and Roberts, G.O. and Gilks, W.R. , Title =. Bayesian Statistics 5 , Editor =

  15. [23]

    Roberts, G. O. and Gelman, A. and Gilks, W. R. , Journal =. Weak convergence and optimal scaling of random walk

  16. [24]

    Kass, R. E. and Carlin, B. P. and Gelman, A. and Neal, R. M. , Journal =

  17. [25]

    , Journal =

    Stephens, M. , Journal =. Bayesian analysis of mixture models with an unknown number of components--an alternative to reversible jump methods , Year =

  18. [26]

    , Journal =

    Godsill, S. , Journal =. On the relationship between

  19. [27]

    Green, P. J. and Mira, A. , Journal =. Delayed rejection in reversible jump

  20. [28]

    and Wong, W

    Liang, F. and Wong, W. H. , Journal =. Real parameter evolutionary

  21. [29]

    and Forster, J

    Dellaportas, P. and Forster, J. J. and Ntzoufras, I. , Journal =. On

  22. [30]

    and Rosenthal, J.S

    Roberts, G. and Rosenthal, J.S. , title =. J. R. Statist. Soc. B. , volume =

  23. [31]

    Brooks, S. P. and Guidici, P. and Roberts, G. O. , Journal =. Efficient construction of reversible jump

  24. [32]

    and Robert, C

    Capp\'e, O. and Robert, C. P. and Ryd\'en, T. , Journal =. Reversible jump

  25. [33]

    and Hurn, M

    Jennison, C. and Hurn, M. A. and Al-Awadhi, F. , Journal =. Discussion of a paper by

  26. [34]

    Green, P. J. , title =. Highly Structured Stochastic Systems. 2003 , publisher =

    2003

  27. [35]

    Sisson, S. A. , Journal =. Transdimensional

  28. [36]

    and Stephens, D

    Jasra, A. and Stephens, D. A. and Holmes, C. C. , Journal =. Population-based reversible jump

  29. [37]

    and Dellaportas, P

    Papathomas, M. and Dellaportas, P. and Vasdekis, V. G. S. , title =

  30. [38]

    and Peters, G

    Fan, Y. and Peters, G. W. and Sisson, S. A. , Journal =. Automating and evaluating reversible jump

  31. [39]

    Brooks, S. P. and Gelman, A. and Jones, G. L. and Meng, X..-L. , title =

  32. [40]

    Neal, R. M. , title =. Handbook of Markov Chain Monte Carlo , editor =. 2011 , type =

    2011

  33. [41]

    and Calderhead, B

    Girolami, M. and Calderhead, B. , title =. J. R. Stat. Soc. B , volume =

  34. [42]

    and Yang, Z

    Rannala, B. and Yang, Z. , Journal =. Improved reversible jump algorithms for

  35. [43]

    Farr, W. M. and Mandel, I. and Stevens, D. , Journal =. An efficient interpolation technique for jump proposals in reversible-jump

  36. [44]

    , Title =

    Yang, Ziheng and Rodriguez, Carlos E. , Title =. Proc. Natl. Acad. Sci. U.S.A. , Volume =

  37. [45]

    and Yang, Z

    Rannala, B. and Yang, Z. , Title =. J. Mol. Evol. , Volume =

  38. [46]

    and Rannala, B

    Yang, Z. and Rannala, B. , Title =. Mol. Biol. Evol. , Volume =

  39. [47]

    and Newton, M.A

    Mau, B. and Newton, M.A. , Title =. J. Computat. Graph. Stat. , Volume =

  40. [48]

    and Pearl, D

    Li, S. and Pearl, D. and Doss, H. , Title =. J. Amer. Statist. Assoc. , Volume =

  41. [49]

    and Teslenko, M

    Ronquist, F. and Teslenko, M. and van der Mark, P. and Ayres, D. L. and Darling, A. and Hohna, S. and Larget, B. and Liu, L. and Suchard, M. A. and Huelsenbeck, J. P. , Title =. Syst. Biol. , Volume =

  42. [50]

    and Lepage, T

    Lartillot, N. and Lepage, T. and Blanquart, S. , title =. Bioinformatics , volume =

  43. [51]

    and Heled, J

    Bouckaert, R. and Heled, J. and Kuhnert, D. and Vaughan, T. and Wu, C. H. and Xie, D. and Suchard, M. A. and Rambaut, A. and Drummond, A. J. , Title =. PLoS Comput. Biol. , Volume =

  44. [52]

    and Lemey, P

    Baele, G. and Lemey, P. and Rambaut, A. and Suchard, M. A. , title =. Bioinformatics , volume =. doi:10.1093/bioinformatics/btx088 , year =

    doi

  45. [53]

    and Simon, D.L

    Larget, B. and Simon, D.L. , title =. Mol. Biol. Evol. , volume =

  46. [54]

    and van der Mark, P

    Lakner, Clemens. and van der Mark, P. and Huelsenbeck, J. P. and Larget, B. and Ronquist, F. , Title =. Syst. Biol. , Volume =

  47. [55]

    and Drummond, Alexei J

    Hohna, Sebastian and Defoin-Platel, M. and Drummond, Alexei J. , Title =. 8th IEEE International Conference on BioInformatics and BioEngineering , Pages =. 2008 , Organization =

    2008

  48. [56]

    , title =

    Hohna, Sebastian and Drummond, Alexei J. , title =. Syst. Biol. , volume =

  49. [57]

    Brown, J. M. and Thomson, R. C. , title =. Syst. Biol. , volume =. doi:10.1093/sysbio/syy008 , year =

    doi

  50. [58]

    and Matsen, F

    Whidden, C. and Matsen, F. A. th , title =. Syst. Biol. , volume =. doi:10.1093/sysbio/syv006 , year =

    doi

  51. [59]

    Chen, Ming-Hui and Kuo, Lynn and Lewis, Paul , Title =

  52. [60]

    and Olsen, G.J

    Swofford, D.L. and Olsen, G.J. and Waddell, P.J. and Hillis, D.M. , Title =. Molecular Systematics , Editor =

  53. [61]

    , Title =

    Felsenstein, J. , Title =. J. Mol. Evol. , Volume =

  54. [62]

    , Title =

    Felsenstein, J. , Title =

  55. [63]

    and Yoder, A.D

    Yang, Z. and Yoder, A.D. , Title =. Syst. Biol. , Volume =

  56. [64]

    Nascimento, F. F. and dos Reis, Mario and Yang, Ziheng , Title =. Nature Ecol. Evol. , Volume =

  57. [65]

    and Yang, Z

    Rannala, B. and Yang, Z. , Title =. Syst. Biol. , Volume =

  58. [66]

    and Zhu, T

    Yang, Z. and Zhu, T. , title =. Proc. Natl. Acad. Sci. U.S.A. , volume =. doi:10.1073/pnas.1712673115 , year =

    doi

  59. [67]

    , Title =

    Yang, Z. , Title =. Mol. Biol. Evol. , Volume =

  60. [68]

    and Jiao, X

    Flouri, T. and Jiao, X. and Rannala, B. and Yang, Z. , Title =. Mol. Biol. Evol. , Volume =

  61. [69]

    and Dalquen, D.A

    Thawornwattana, Y. and Dalquen, D.A. and Yang, Z. , title =. Bayesian Analysis , volume =. doi:10.1214/17-BA1084 , year =

    doi

  62. [70]

    Zanella, Giacomo , title =. J. Am. Statist. Ass. , volume =

  63. [71]

    Thomson, R. C. and Brown, J. M. , title =. Syst. Biol. , volume =. doi:10.1093/sysbio/syac002 , year =

    doi

  64. [72]

    and Huelsenbeck, J

    Zhang, C. and Huelsenbeck, J. P. and Ronquist, F. , title =. Syst. Biol. , volume =. doi:10.1093/sysbio/syaa002 , year =

    doi

  65. [73]

    , title =

    Meyer, X. , title =. Syst. Biol. , volume =. doi:10.1093/sysbio/syab004 , year =

    doi

  66. [74]

    H., Xie, D., Suchard, M

    Bouckaert, R., Heled, J., Kuhnert, D., Vaughan, T., Wu, C. H., Xie, D., Suchard, M. A., Rambaut, A., and Drummond, A. J. 2014. BEAST 2: a software platform for Bayesian evolutionary analysis. PLoS Comput. Biol. , 10 (4): e1003537

    2014

  67. [75]

    P., Guidici, P., and Roberts, G

    Brooks, S. P., Guidici, P., and Roberts, G. O. 2003. Efficient construction of reversible jump Markov chain Monte Carlo proposal distributions. J. R. Stat. Soc. B\/ , 65 : 3--39

    2003

  68. [76]

    P., and Ryd\'en, T

    Capp\'e, O., Robert, C. P., and Ryd\'en, T. 2003. Reversible jump MCMC converging to birth-and-death MCMC and more general continuous time samplers. J. R. Stat. Soc. B\/ , 65 : 679--700

    2003

  69. [77]

    Carlin, B. P. and Chib, S. 1995. Bayesian model choice via Markov chain Monte Carlo . J. R. Stat. Soc. B\/ , 57 : 473--484

    1995

  70. [78]

    Chen, M.-H., Kuo, L., and Lewis, P. 2014. Bayesian Phylogenetics: Methods, Algorithms, and Applications\/ . Chapman and Hall/CRC, London

    2014

  71. [79]

    J., and Ntzoufras, I

    Dellaportas, P., Forster, J. J., and Ntzoufras, I. 2002. On Bayesian model and variable selection using MCMC . Stat. Comput. , 12 : 27--36

    2002

  72. [80]

    Felsenstein, J. 2004. Inferring Phylogenies\/ . Sinauer Associates, Sunderland, Massachusetts

    2004

  73. [81]

    Flouri, T., Jiao, X., Rannala, B., and Yang, Z. 2018. Species tree inference with BPP using genomic sequences and the multispecies coalescent. Mol. Biol. Evol. , 35 (10): 2585--2593

    2018

  74. [82]

    Flouri, T., Jiao, X., Rannala, B., and Yang, Z. 2020. A Bayesian implementation of the multispecies coalescent model with introgression for phylogenomic analysis. Mol. Biol. Evol. , 37 (4): 1211--1223

    2020

  75. [83]

    Frigessi, A., Hwang, C.-R., and Younes, L. 1992. Optimal spectral structure of reversible stochastic matrices, Monte Carlo methods and the simulation of Markov random fields. Ann. Appl. Prob. , 2 (3): 610--628

    1992

  76. [84]

    Gelman, A., Roberts, G., and Gilks, W. 1996. Efficient Metropolis jumping rules. In J. Bernardo, J. Berger, A. Dawid, and A. Smith, editors, Bayesian Statistics 5\/ , volume 5, pages 599--607. Oxford University Press, Oxford

    1996

  77. [85]

    Geyer, C. J. 1992. Practical Markov chain Monte Carlo . Stat. Sci. , 7 : 473--483

    1992

  78. [86]

    and Calderhead, B

    Girolami, M. and Calderhead, B. 2011. Riemann manifold Langevin and Hamiltonian Monte Carlo methods. J. R. Stat. Soc. B\/ , 73 (2): 123--214

    2011

  79. [87]

    Godsill, S. 2001. On the relationship between Markov chain Monte Carlo methods for model uncertainty. J. Comput. Graph. Stat. , 10 : 1--19

    2001

  80. [88]

    Green, P. J. 1995. Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika\/ , 82 : 711--732

    1995

  81. [89]

    Green, P. J. and Han, X. L. 1992. Metropolis methods, Gaussian proposals and antithetic variables. In P. Barone, A. Frigessi, and M. Piccioni, editors, Stochastic Models, Statistical Methods and Algorithms in Image Analysis\/ , pages 142--164. Springer, New York

    1992

  82. [90]

    and Miller, M

    Grenander, U. and Miller, M. I. 1994. Representations of knowledge in complex systems. J. R. Stat. Soc. B\/ , 56 : 549--603

    1994

  83. [91]

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

    1970

  84. [92]

    and Drummond, A

    Hohna, S. and Drummond, A. J. 2012. Guided tree topology proposals for Bayesian phylogenetic inference. Syst. Biol. , 61 : 1--11

    2012

  85. [93]

    Hohna, S., Defoin-Platel, M., and Drummond, A. J. 2008. Clock-constrained tree proposal operators in Bayesian phylogenetic inference. In 8th IEEE International Conference on BioInformatics and BioEngineering\/ , page 10.1109/BIBE.2008.4696663, Athens (Greece). IEEE

    2008 doi

  86. [94]

    A., and Holmes, C

    Jasra, A., Stephens, D. A., and Holmes, C. C. 2007. Population-based reversible jump Markov chain Monte Carlo . Biometrika\/ , 94 : 787--807

    2007

  87. [95]

    A., and Al-Awadhi, F

    Jennison, C., Hurn, M. A., and Al-Awadhi, F. 2003. Discussion of a paper by Brooks et al. J. R. Stat. Soc. B\/ , 65 : 44--45

    2003

  88. [96]

    and Cantor, C

    Jukes, T. and Cantor, C. 1969. Evolution of protein molecules. In Munro, H.N. , ed. Mammalian Protein Metabolism\/ , pages 21--123. Academic Press, New York

    1969

  89. [97]

    and Snell, J

    Kemeny, J. and Snell, J. 1960. Finite Markov Chains\/ . Princeton, Van Nostrand

    1960

  90. [98]

    P., Larget, B., and Ronquist, F

    Lakner, C., van der Mark, P., Huelsenbeck, J. P., Larget, B., and Ronquist, F. 2008. Efficiency of Markov chain Monte Carlo tree proposals in Bayesian phylogenetics. Syst. Biol. , 57 : 86--103

    2008

  91. [99]

    and Simon, D

    Larget, B. and Simon, D. 1999. Markov chain Monte Carlo algorithms for the Bayesian analysis of phylogenetic trees. Mol. Biol. Evol. , 16 : 750--759

    1999

  92. [100]

    Lartillot, N., Lepage, T., and Blanquart, S. 2009. PhyloBayes 3: a Bayesian software package for phylogenetic reconstruction and molecular dating. Bioinformatics\/ , 25 (17): 2286--2288

    2009

  93. [101]

    Li, S., Pearl, D., and Doss, H. 2000. Phylogenetic tree reconstruction using Markov chain Monte Carlo . J. Amer. Statist. Assoc. , 95 : 493--508

    2000

  94. [102]

    Lindley, D. 1957. A statistical paradox. Biometrika\/ , 44 : 187--192

    1957

  95. [103]

    and Newton, M

    Mau, B. and Newton, M. 1997. Phylogenetic inference for binary data on dendrograms using Markov chain Monte Carlo . J. Computat. Graph. Stat. , 6 : 122--131

    1997

  96. [104]

    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. J. Chem. Phys. , 21 : 1087--1092

    1953

  97. [105]

    Meyer, X. 2021. Adaptive tree proposals for Bayesian phylogenetic inference. Syst. Biol. , 70 (5): 1015--1032

    2021

  98. [106]

    M., Slighton, J

    Miyamoto, M. M., Slighton, J. L., and Goodman, M. 1987. Phylogenetic relations of humans and african apes from DNA sequences in the psi-eta-globin region. Science\/ , 238 : 369--373

    1987

  99. [107]

    F., dos Reis, M., and Yang, Z

    Nascimento, F. F., dos Reis, M., and Yang, Z. 2017. A biologist's guide to Bayesian phylogenetic analysis. Nature Ecol. Evol. , 1 : 1446--1454

    2017

  100. [108]

    Neal, R. M. 2011. MCMC using Hamiltonian dynamics. In S. Brooks, A. Gelman, G. Jones, and X.-L. Meng, editors, Handbook of Markov Chain Monte Carlo\/ , pages 113--162. Chapman and Hall/CRC, London

    2011

  101. [109]

    Nishimura, A., Dunson, D., and Lu, J. 2020. Discontinuous H amiltonian M onte C arlo for discrete parameters and discontinuous likelihoods. Biometrika\/ , 107 (2): 365--380

    2020

  102. [110]

    Peskun, P. 1973. Optimum Monte-Carlo sampling using Markov chains. Biometrika\/ , 60 (3): 607--612

    1973

  103. [111]

    and Yang, Z

    Rannala, B. and Yang, Z. 1996. Probability distribution of molecular evolutionary trees: a new method of phylogenetic inference. J. Mol. Evol. , 43 : 304--311

    1996

  104. [112]

    and Yang, Z

    Rannala, B. and Yang, Z. 2017. Efficient Bayesian species tree inference under the multispecies coalescent. Syst. Biol. , 66 : 823--842

    2017

  105. [113]

    and Rosenthal, J

    Roberts, G. and Rosenthal, J. 2008. Optimal scaling of discrete approximations to Langevin diffusions. J. R. Statist. Soc. B. , 60 (255--226)

    2008

  106. [114]

    L., Darling, A., Hohna, S., Larget, B., Liu, L., Suchard, M

    Ronquist, F., Teslenko, M., van der Mark, P., Ayres, D. L., Darling, A., Hohna, S., Larget, B., Liu, L., Suchard, M. A., and Huelsenbeck, J. P. 2012. MrBayes 3.2: Efficient Bayesian phylogenetic inference and model choice across a large model space. Syst. Biol. , 61 : 539--542

    2012

  107. [115]

    Sokal, A. 1989. Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms\/ . Lecture Notes for the Cours de Troisieme Cycle de la Physique en Suisse Romande, Lausanne, Switzerland (June 1989)

    1989

  108. [116]

    Stephens, M. 2000. Bayesian analysis of mixture models with an unknown number of components--an alternative to reversible jump methods. Ann. Stat. , 28 : 40--74

    2000

  109. [117]

    Swofford, D., Olsen, G., Waddell, P., and Hillis, D. 1996. Phylogeny inference. In D. M. Hillis, C. Moritz, and B. K. Mable, editors, Molecular Systematics\/ , pages 407--514. Sinauer Associates, Sunderland, Massachusetts, 2 edition

    1996

  110. [118]

    Thawornwattana, Y., Dalquen, D., and Yang, Z. 2018. Designing simple and efficient Markov chain Monte Carlo proposal kernels. Bayesian Analysis\/ , 13 (4): 1033--1059

    2018

  111. [119]

    Thomson, R. C. and Brown, J. M. 2022. On the need for new measures of phylogenomic support. Syst. Biol. , 71 : 917--920

    2022

  112. [120]

    and Matsen, F

    Whidden, C. and Matsen, F. A. t. 2015. Quantifying MCMC exploration of phylogenetic tree space. Syst. Biol. , 64 (3): 472--491

    2015

  113. [121]

    Yang, Z. 2014. Molecular Evolution: A Statistical Approach\/ . Oxford University Press, Oxford, UK

    2014

  114. [122]

    and Rannala, B

    Yang, Z. and Rannala, B. 1997. Bayesian phylogenetic inference using DNA sequences: a Markov chain Monte Carlo method. Mol. Biol. Evol. , 14 : 717--724

    1997

  115. [123]

    and Rodriguez, C

    Yang, Z. and Rodriguez, C. E. 2013. Searching for efficient Markov chain Monte Carlo proposal kernels. Proc. Natl. Acad. Sci. U.S.A. , 110 (48): 19307--19312

    2013

  116. [124]

    and Yoder, A

    Yang, Z. and Yoder, A. 2003. Comparison of likelihood and Bayesian methods for estimating divergence times using multiple gene loci and calibration points, with application to a radiation of cute-looking mouse lemur species. Syst. Biol. , 52 : 705--716

    2003

  117. [125]

    and Zhu, T

    Yang, Z. and Zhu, T. 2018. Bayesian selection of misspecified models is overconfident and may cause spurious posterior probabilities for phylogenetic trees. Proc. Natl. Acad. Sci. U.S.A. , 115 (8): 1854--1859

    2018

  118. [126]

    Zanella, G. 2020. Informed proposals for local MCMC in discrete spaces. J. Am. Statist. Ass. , 115 (530): 852--865

    2020

  119. [127]

    P., and Ronquist, F

    Zhang, C., Huelsenbeck, J. P., and Ronquist, F. 2020. Using parsimony-guided tree proposals to accelerate convergence in Bayesian phylogenetic inference. Syst. Biol. , 69 (5): 1016--1032

    2020

Pith tools

Reviewed July 9, 2026 · model on record in the stance chip above.