Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

Large sample scaling analysis of the Zig-Zag algorithm for Bayesian inference

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read With control variates, a sub-sampled Zig-Zag sampler can draw independent posterior samples at O(1) cost as the data size grows.

desk verdict First data-size scaling analysis for PDMPs with solid limit theorems, but the headline O(1) complexity claim rests on vacuous corollaries that start at the target distribution. read the letter →

arxiv 2411.14983 v2 pith:3FNDUZJ5 submitted 2024-11-22 stat.CO

Sanket Agrawal , Joris Bierkens , Gareth O. Roberts This is my paper · ORCID
classification stat.CO MSC 60F1760J2562F1562E2065C05
keywords piecewisedeterministicMarkovprocessesZig-Zagsamplersub-samplingcontrolvariatesfluidlimitslarge-sampleasymptoticsBayesiancomputationcomputationalcomplexity
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 asks what happens to the Zig-Zag sampler, a Markov chain Monte Carlo method that moves at constant speed and randomly flips direction, when the number of data points $n$ grows. In the stationary phase it proves that the control-variate sub-sampling version (ZZ-CV) needs only $O(n^{-1/2})$ time and $O(1)$ proposed velocity switches per essentially independent sample, with each switch costing $O(1)$, so the total work per sample is $O(1)$. That is an $O(n)$ speed-up over canonical Zig-Zag and over traditional MCMC, both of which cost $O(n)$. In the transient phase the paper shows all Zig-Zag variants are approximated by a deterministic flow that drifts toward decreasing KL divergence, with sub-sampling damping the drift. These are the first scaling limits for a piecewise deterministic sampler as a function of data size rather than dimension.

What carries the argument

The key mechanism is the imbalance between switching rates in opposite velocity directions, which produces drift, together with the magnitude of the total switching rate. In the transient phase the drift $b_i(x) = -E[S_i(x;Y)]/(\lambda_i(x,-\mathbf{1})+\lambda_i(x,\mathbf{1}))$ defines a fluid-limit ODE, with the denominator acting as the damping factor from sub-sampling. In the stationary phase the rescaled coordinate $\xi = n^{1/2}(x - \hat{x}_n)$ separates the regimes: ZZ-SS has switching rates of order $n$, leading to a diffusion limit, while ZZ-CV and canonical Zig-Zag have effective rates of order $n^{1/2}$ after time rescaling, leading to Zig-Zag limits. The complexity table then multiplies time-to-mixing, switching rate, and per-switch cost to obtain the $O(1)$ versus $O(n)$ totals.

What would settle it

Run ZZ-CV in the stationary phase with a reference point whose error is $O_p(n^{-\alpha})$ for $\alpha < 1/2$ (for instance, an estimator based on $n^{1/3}$ observations) and measure the total CPU time per effectively independent sample; if the time grows with $n$ rather than remaining bounded, the $O(1)$ claim fails. A more direct check is to count proposed switches per independent sample and confirm that they grow like $n^{1-2\alpha}$ instead of staying $O(1)$ in this regime.

Watch

Extended reading notes

Core claim

In the stationary phase, after rescaling coordinates by the posterior contraction rate, the paper proves weak convergence of the Zig-Zag process in three regimes. Vanilla sub-sampling (ZZ-SS) converges to an Ornstein-Uhlenbeck diffusion with a damping matrix determined by the subsample estimator, and it needs $O(1)$ time but $O(n)$ switching events, so its total cost is $O(n)$. Canonical Zig-Zag and ZZ-CV, slowed down by a factor $n^{1/2}$, converge to Zig-Zag processes invariant for the limiting Gaussian posterior. Since ZZ-CV estimates the gradient from random batches of fixed size, each proposed switch costs $O(1)$, and the process reaches an essentially independent sample in $O(n^{-1/2})$ time with $O(1)$ proposed switches, giving $O(1)$ total effort. This $O(1)$ claim holds only if the reference point used by the control variates satisfies $\|X^*_n - \hat{x}_n\| = O_p(n^{-1/2})$; otherwise the switching rates grow to $O(n)$ and ZZ-CV degrades to the slow diffusive regime.

Load-bearing premise

The $O(1)$ cost of ZZ-CV rests on the assumption that the reference point used for control variates is within $O_p(n^{-1/2})$ of the posterior mode; if the reference point is farther away, the switching rates grow to $O(n)$ and the advantage disappears.

Editorial extensions

If this is right

  • ZZ-CV with a good reference point obtains an essentially independent sample in $O(1)$ total work as $n \to \infty$, matching the 'superefficiency' claim of the original Zig-Zag paper.
  • Canonical Zig-Zag and ZZ-SS both have $O(n)$ total cost per independent sample; the gain from control variates is realized only when the reference point is already within $O_p(n^{-1/2})$ of the posterior mode.
  • In the transient phase, canonical Zig-Zag converges to the minimizer at optimal speed $\pm 1$, while ZZ-SS slows down near the mode and ZZ-CV can be arbitrarily slow in heavy-tailed models.
  • A mixed scheme that uses ZZ-SS in the tails and ZZ-CV near the mode converges faster than either pure scheme in heavy-tailed examples.
  • Under the Bernstein-von Mises theorem, the stationary limits are invariant for the asymptotic Gaussian posterior, so the scaling analysis transfers directly to the standard large-sample Bayesian setting.

Reading between the lines

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

  • The same scaling logic suggests that updating the reference point adaptively as the sampler runs could preserve the $O(1)$ complexity, provided the update error stays $O_p(n^{-1/2})$; the paper advocates this but does not analyze it.
  • The fluid-limit result is not special to Bayesian settings: the paper notes that the ODE approximation holds whenever switching rates grow as $n^{\beta}$ with position speed $n^{\alpha}$, so the transient analysis should transfer to other piecewise deterministic samplers such as the bouncy particle sampler.
  • For heavy-tailed models, Corollary 5.1 implies a mixed scheme that provably beats both pure schemes in the transient phase; testing this on real heavy-tailed posteriors is a natural next step.
  • Because the analysis fixes dimension $d$, an open question is how the $O(1)$ complexity interacts with dimension; if the per-switch cost or the reference-point error degrades with $d$, the practical speed-up may be smaller than the headline $O(n)$ factor.
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 / 4 minor

Summary. The paper studies the large-sample (n → ∞) behavior of the Zig-Zag sampler and its sub-sampling variants, ZZ-SS and ZZ-CV, in parametric Bayesian inference with fixed dimension. In the transient phase, the authors prove a fluid limit: Zig-Zag trajectories converge to the solution of the ODE x'(t) = b(x) with drift (3.3) given by the negative expected score normalized by the local switching rate. In the stationary phase, they derive three different limits under the n^{1/2} rescaling: ZZ-SS converges to an Ornstein-Uhlenbeck diffusion (4.4), while slowed-down ZZ-CV and canonical Zig-Zag converge to Zig-Zag processes with rates (4.6) and the canonical rate, respectively. From these limits, the paper claims complexity bounds per essentially independent sample: O(n) for canonical Zig-Zag and ZZ-SS, and O(1) for ZZ-CV (Table 1). The proofs are detailed and self-contained in the appendix, and the limiting objects are internally consistent: for example, the OU drift in Theorem 4.2 preserves the asymptotic posterior N(0, I(x0)^{-1}), and the limiting Zig-Zag rate in Remark 4.7 has the correct skew-symmetric structure for that Gaussian target.

Significance. If the complexity conclusions are valid, the paper is a substantial contribution: it is the first scaling-limit analysis of PDMP samplers as a function of dataset size, it covers misspecified models, and it provides explicit and checkable limiting equations (3.3), (4.4), and (4.6) that explain both the transient optimization-like behavior and the stationary mixing differences between sub-sampling schemes. The strength of the paper is its careful martingale and averaging arguments, which appear to support the qualitative distinction between ZZ-SS (diffusive, O(n) effort) and ZZ-CV (Zig-Zag limit, potentially O(1) effort). However, the headline complexity claim, and specifically the O(n^{-1/2}) time-to-independent-sample for ZZ-CV, is currently supported by Corollary 4.12, and that corollary is vacuous as stated. Because this is the central claim advertised in the abstract and Table 1, the paper requires substantial revision before the complexity contribution can be accepted.

major comments (3)
  1. [§4.3, Corollary 4.12 (and Corollary 4.5)] Corollary 4.12 is vacuous as stated and cannot support the O(n^{-1/2}) mixing-time claim. The corollary takes U^n_0 ∼ Π^(n)_♯ξ, i.e. the initial law is exactly the target in the rescaled coordinate. Since each Zig-Zag process is Π^(n)-invariant, L(ξ^n_{n^{-1/2}t}) = Π^(n)_♯ξ for all t ≥ 0, so the KR-distance bound holds trivially with T = 0. A trivial bound with T = 0 cannot imply that the algorithm needs O(n^{-1/2}) time to obtain an essentially independent sample from a non-stationary initial distribution. The same objection applies to Corollary 4.5, which also starts at Π^(n)_♯ξ. The complexity rows in Table 1 and the claims in Section 4.3 therefore rest on a statement that proves nothing about mixing from a general starting point. The authors need either to prove the mixing-time claim from a non-stationary initial distribution or to explicitly restate and limit the contribution as a stationary-phase cost-per-sample result rather than a time-to-independent-sample result.
  2. [§4.2, Proof of Corollary 4.12] Even if the initial distribution were non-stationary, the proof sketch ('same as Corollary 4.5 ... with a slight modification of Theorem 1 of [44]') does not supply the missing hypotheses. Theorem 1 of Roberts and Rosenthal (2016) gives complexity bounds for diffusion limits under conditions that include exponential ergodicity of the limiting process in the relevant metric and a transfer of that ergodicity to the prelimit processes uniformly in n. The paper does not verify exponential ergodicity of the limiting Zig-Zag process with rates (4.6) in the KR metric, nor does it establish the uniform-in-n bound linking Theorem 4.6 to a finite-time KR error. Weak convergence alone does not provide a rate of convergence, and the uniform-in-n statement in the corollary is precisely what is needed for the complexity calculation. This is a load-bearing gap, not a cosmetic one.
  3. [§4.2, Remark 4.10 and Theorem 4.6] The O(1) complexity conclusion is conditional on an assumption about the reference point that is not guaranteed by the algorithm. Theorem 4.6 requires ξ^*_n → ξ^* in probability, which in the notation of Section 4.2 means ∥X^*_n − x̂_n∥ = O_p(n^{-1/2}) with a nontrivial limit. If the reference point is only O_p(1) away, Remark 4.10 shows that the ZZ-CV switching rates are O(n) and the process reverts to the diffusive ZZ-SS regime with O(n) total cost. The paper does acknowledge a one-off O(n) initialization cost to find a good reference point, but the theorem itself does not provide a constructive or guaranteed mechanism for selecting T_n satisfying the required rate. The authors should state this condition explicitly as a hypothesis of the complexity theorem and discuss whether it is verifiable in practice, rather than treating it as an automatic consequence of the ZZ-CV construction.
minor comments (4)
  1. [§3, equation (3.3)] Equation (3.3) is defined on R^d \ H(ω), but the final clause says 'x ∈ H(ω)'; this should read 'x ∉ H(ω)'.
  2. [§4.1, Corollary 4.5 proof] The proof of Corollary 4.5 says 'combining this and Theorem 4.6', but the relevant result is Theorem 4.2; Corollary 4.12 similarly refers to 'Theorem 4.2 or 4.11' where it should refer to Theorem 4.6 or 4.11.
  3. [§2.5 and §4.1] The notation for the subsample collection is inconsistent: S(n,m) and S(m,n) are both used, and at one point S(m,n) is written where S(n,m) appears intended; please standardize.
  4. [Abstract and Section 1] Minor spelling issue: 'Kullback-Liebler' should be 'Kullback-Leibler'.

Circularity Check

1 steps flagged · score 6.0 of 10

Stationary-phase mixing claim rests on a tautological corollary: the Zig-Zag process is started at the target posterior, so the KR bound is identically zero.

  1. self definitional [Section 4.1, Corollary 4.5; Section 4.2, Corollary 4.12; used in Section 4.3 and Table 1]
    "Corollary 4.12. Let (U n t )t≥0 be as in Theorem 4.6 or 4.11 and suppose the technical assumptions are satisfied. Then, for any ϵ >0, there are δ >0, T <∞, and N <∞, such that, P EU n 0 ∼Π(n) ♯ξ L(ξ n n−1/2t) − Π(n) ♯ξ KR < ϵfor all t ≥ T, n≥ N > 1 − δ."

    Corollary 4.12 initializes the Zig-Zag process at the target posterior Π^(n)_♯ξ. Because the Zig-Zag rates (2.3)/(2.6) are constructed so that Π^(n) is invariant (Section 2.2), the marginal law of ξ^n_{n^{-1/2}t} equals Π^(n)_♯ξ for every t≥0 and n. Hence the KR distance is identically 0 and the corollary is true for T=0, without invoking Theorem 4.6 or the diffusion-limit result [44]. Section 4.3 nevertheless uses Corollary 4.12 to conclude that canonical Zig-Zag and ZZ-CV 'require O(n^{-1/2}) time to obtain an essentially independent sample'; Table 1 repeats this. That time-to-sample claim is not derived from the weak-convergence theorems; it is put into the statement by starting at stationarity. Corollary 4.5 for ZZ-SS is identical in structure.

full rationale

The scaling limits themselves (Theorems 3.3, 4.2, 4.6, 4.11) are derived from the stated statistical Assumptions 2.1–2.4 and U-statistic laws of large numbers; those derivations are self-contained and are not circular. The arithmetic converting event rates and per-event costs into total cost is also not circular. The circularity is confined to the corollaries that turn these limits into 'time to an essentially independent sample': by initializing at the target distribution Π^(n)_♯ξ, the KR bound is satisfied trivially at all times, so the corollary contains no mixing-time information. Removing that support leaves the Table 1 complexity entries and the abstract's O(1) ZZ-CV claim unsupported as written, although they may be repairable with a nonstationary initial law and a careful application of an external diffusion-limit theorem. No fitted parameter is renamed as a prediction, and the self-citations to [14] and [44] are not the source of circularity in the weak-convergence proofs.

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

The central results rest on the stated statistical assumptions (smoothness, moments, unique KL minimizer, Bernstein-von Mises) and on the reference-point conditions. No free parameters are fitted to data. The assumptions are standard for large-sample Bayesian asymptotics, though the independence of ξ* in Theorem 4.6 is questionable for data-dependent reference points.

assumptions (6)
  • domain assumption Assumption 2.1: For each y, the log-density is C^3 in x and the Hessian of the score function is bounded by M'I.
    Imposes smoothness needed for the Taylor expansions in the fluid limit proofs.
  • domain assumption Assumption 2.2: First moments of the score and its derivative exist under P.
    Used for laws of large numbers and uniform convergence arguments.
  • domain assumption Assumption 2.3: The KL divergence is uniquely minimized at x0.
    Defines the point of posterior concentration and the direction of the fluid drift.
  • domain assumption Assumption 2.4: There exist estimators x̂_n solving the score equation with n^{-1/2} posterior contraction and Bernstein-von Mises with total variation convergence.
    Ensures the posterior contracts at the n^{-1/2} rate which defines the stationary-phase scaling.
  • domain assumption Assumption 3.1: Reference points X*_n converge almost surely to X* independent of the data with finite second moment.
    Used to derive the asymptotic drift for ZZ-CV in the transient phase; questionable for data-dependent reference points.
  • domain assumption Before Theorem 4.6: reference points satisfy ||X*_n - x̂_n|| = O_p(n^{-1/2}) and ξ* is independent of the data.
    Load-bearing for the O(1) complexity of ZZ-CV; without it, the rates are O(n) and the diffusive scaling applies (Remark 4.10).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Large sample scaling analysis of the Zig-Zag algorithm for Bayesian inference." pith.science (2026). https://pith.science/paper/3FNDUZJ5

@misc{pith2026241114983,
  author       = {Pith},
  title        = {Pith review of: Large sample scaling analysis of the Zig-Zag algorithm for Bayesian inference},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3FNDUZJ5}},
  note         = {Machine review of arXiv:2411.14983}
}
read the original abstract

Piecewise deterministic Markov processes provide scalable methods for sampling from the posterior distributions in big data settings by admitting principled sub-sampling strategies that do not bias the output. An important example is the Zig-Zag process of [Ann. Stats. 47 (2019) 1288 - 1320] where clever sub-sampling has been shown to produce an essentially independent sample at a cost that does not scale with the size of the data. However, sub-sampling also leads to slower convergence and poor mixing of the process, a behaviour which questions the promised scalability of the algorithm. We provide a large sample scaling analysis of the Zig-Zag process and its sub-sampling versions in settings of parametric Bayesian inference. In the transient phase of the algorithm, we show that the Zig-Zag trajectories are well approximated by the solution to a system of ODEs. These ODEs possess a drift in the direction of decreasing KL-divergence between the assumed model and the true distribution and are explicitly characterized in the paper. In the stationary phase, we give weak convergence results for different versions of the Zig-Zag process. Based on our results, we estimate that for large data sets of size n, using suitable control variates with sub-sampling in Zig-Zag, the algorithm costs O(1) to obtain an essentially independent sample; a computational speed-up of O(n) over the canonical version of Zig-Zag and other traditional MCMC methods

Figures

Figures reproduced from arXiv: 2411.14983 by the authors.

Figure 1
Figure 1. Trajectories for different versions of the Zig-Zag algorithm targeting a Bayesian poste [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Trajectories of the canonical Zig-Zag process in 2 dimensions with different starting values. The dashed lines represent the hyper￾surfaces Hn i , i = 1, 2. Paths indicated in black stick to one of the hypersurfaces as soon as they hit it whereas those in orange don’t. Colours were coded after the paths were observed [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 4
Figure 4. Stationary phase trajectories of ZZ-CV for Bayesian logistic regression example with [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figures from the paper (3 more)
Figure 5
Figure 5. Figure 5: Asymptotic drift for different models and sub-sampling schemes in one dimension with [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 6
Figure 6. Figure 6: Bayesian Logistic regression example with 10000 data points. The black curves are [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 7
Figure 7. Figure 7: Trajectories of Zig-Zag process for different sub-sampling schemes targeting posterior [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Scaling Results for Piecewise Deterministic Monte Carlo : A Survey

    stat.CO 2026-07 conditional novelty 1.0 of 10

    The Zig-Zag and Bouncy Particle samplers have known dimension, anisotropy, and big-data scaling limits that make their efficiency costs predictable, and this survey assembles them with tuning heuristics.

Reference graph

Works this paper leans on

52 extracted references · 51 canonical work pages · cited by 1 Pith paper

  1. [44]

    Roberts, G. O. and Rosenthal, J. S. (2016). Complexity bounds for Markov chain Monte Carlo algorithms via diffusion limits. Journal of Applied Probability , 53(2):410–420

    work page 2016

  2. [1]

    Altmeyer, R. (2022). Polynomial time guarantees for sampling based posterior inference in high-dimensional generalised linear models. arXiv preprint arXiv:2208.13296 . 47

    work page Pith review arXiv 2022

  3. [2]

    Andrieu, C., Durmus, A., N¨ usken, N., and Roussel, J. (2021). Hypocoercivity of piecewise deterministic markov process-monte carlo. The Annals of Applied Probability , 31(5):2478– 2517

    work page 2021

  4. [3]

    and Livingstone, S

    Andrieu, C. and Livingstone, S. (2021). Peskun–tierney ordering for markovian monte carlo: beyond the reversible scenario. The Annals of Statistics , 49(4):1958–1981

    work page 2021

  5. [4]

    Arcones, M. A. and Gin´ e, E. (1993). Limit theorems for U-processes. The Annals of Proba- bility, pages 1494–1542

    work page 1993

  6. [5]

    O., and Zanella, G

    Ascolani, F., Roberts, G. O., and Zanella, G. (2024). Scalability of Metropolis-within-Gibbs schemes for high-dimensional Bayesian models

    work page 2024

  7. [6]

    and Zanella, G

    Ascolani, F. and Zanella, G. (2024). Dimension-free mixing times of Gibbs samplers for Bayesian hierarchical models. The Annals of Statistics , 52(3):869–894. Publisher: Institute of Mathematical Statistics

    work page 2024

  8. [7]

    B., and Nemeth, C

    Baker, J., Fearnhead, P., Fox, E. B., and Nemeth, C. (2019). Control variates for stochastic gradient mcmc. Statistics and Computing , 29:599–615

    work page 2019

Show all 52 references
  1. [8]

    Bardenet, R., Doucet, A., and Holmes, C. (2017). On markov chain monte carlo methods for tall data. Journal of Machine Learning Research , 18(47):1–43

    2017

  2. [9]

    Berk, R. H. (1970). Consistency a posteriori. The Annals of Mathematical Statistics , 41(3):894–906

    1970

  3. [10]

    and Bierkens, J

    Bertazzi, A. and Bierkens, J. (2022). Adaptive schemes for piecewise deterministic monte carlo algorithms. Bernoulli, 28(4):2404–2430

    2022

  4. [11]

    Bertazzi, A., Bierkens, J., and Dobson, P. (2022). Approximations of piecewise deter- ministic markov processes and their convergence properties. Stochastic Processes and their Applications, 154:91–153

    2022

  5. [12]

    and Duncan, A

    Bierkens, J. and Duncan, A. (2017). Limit theorems for the zig-zag process. Advances in Applied Probability, 49(3):791–825

    2017

  6. [13]

    Bierkens, J., Fearnhead, P., and Roberts, G. O. (2019a). Supplement to “The zig-zag process and super-efficient sampling for Bayesian analysis of big data

    2019

  7. [14]

    Bierkens, J., Fearnhead, P., and Roberts, G. O. (2019b). The zig-zag process and super- efficient sampling for Bayesian analysis of big data. The Annals of Statistics , 47(3):1288–1320

    2019

  8. [15]

    Bierkens, J., Grazzi, S., Kamatani, K., and Roberts, G. (2020). The boomerang sampler. In International conference on machine learning , pages 908–918. PMLR

    2020

  9. [16]

    Bierkens, J., Kamatani, K., and Roberts, G. O. (2022). High-dimensional scaling limits of piecewise deterministic sampling algorithms. The Annals of Applied Probability , 32(5):3361– 3407

    2022

  10. [17]

    Bierkens, J., Kamatani, K., and Roberts, G. O. (2025). Scaling of piecewise deterministic Monte Carlo for anisotropic targets. Bernoulli, 31(3):2323–2350. Publisher: Bernoulli Society for Mathematical Statistics and Probability

    2025

  11. [18]

    Bierkens, J., Nyquist, P., and Schlottke, M. C. (2021). Large deviations for the empirical measure of the zig-zag process. The Annals of Applied Probability , 31(6):2811–2843

    2021

  12. [19]

    O., and Zitt, P.-A

    Bierkens, J., Roberts, G. O., and Zitt, P.-A. (2019c). Ergodicity of the zigzag process. The Annals of Applied Probability , 29(4):2266–2301. 48

    2019

  13. [20]

    J., and Doucet, A

    Bouchard-Cˆ ot´ e, A., Vollmer, S. J., and Doucet, A. (2018). The bouncy particle sampler: A nonreversible rejection-free Markov chain Monte Carlo method. Journal of the American Statistical Association, 113(522):855–867

    2018

  14. [21]

    L., and Rigollet, P

    Chewi, S., Lu, C., Ahn, K., Cheng, X., Gouic, T. L., and Rigollet, P. (2021). Optimal dimension dependence of the Metropolis-Adjusted Langevin Algorithm. In Proceedings of Thirty Fourth Conference on Learning Theory , pages 1260–1300. PMLR. ISSN: 2640-3498

    2021

  15. [22]

    F., Roberts, G

    Christensen, O. F., Roberts, G. O., and Rosenthal, J. S. (2005). Scaling limits for the transient phase of local Metropolis–Hastings algorithms. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 67(2):253–268

    2005

  16. [23]

    Cochran, W. G. (1977). Sampling techniques. John Wiley & Sons

    1977

  17. [24]

    E., and Roberts, G

    Corbella, A., Spencer, S. E., and Roberts, G. O. (2022). Automatic Zig-Zag sampling in practice. arXiv preprint arXiv:2206.11410

    2022 arXiv

  18. [25]

    Cornish, R., Vanetti, P., Bouchard-Cˆ ot´ e, A., Deligiannidis, G., and Doucet, A. (2019). Scalable metropolis-hastings for exact bayesian inference with large datasets. In International Conference on Machine Learning , pages 1351–1360. PMLR

    2019

  19. [26]

    Davis, M. (1993). Markov Models & Optimization . Taylor & Francis

    1993

  20. [27]

    Davis, M. H. (1984). Piecewise-deterministic Markov processes: A general class of non- diffusion stochastic models. Journal of the Royal Statistical Society: Series B (Methodological), 46(3):353–376

    1984

  21. [28]

    Deligiannidis, G., Paulin, D., Bouchard-Cˆ ot´ e, A., and Doucet, A. (2021). Randomized hamiltonian monte carlo as scaling limit of the bouncy particle sampler and dimension-free convergence rates. The Annals of Applied Probability , 31(6):2612–2662

    2021

  22. [29]

    J., and Yu, B

    Dwivedi, R., Chen, Y., Wainwright, M. J., and Yu, B. (2019). Log-concave sampling: Metropolis-Hastings algorithms are fast. Journal of Machine Learning Research , 20(183):1– 42

    2019

  23. [30]

    Ethier, S. N. and Kurtz, T. G. (1986). Markov processes: Characterization and convergence. John Wiley & Sons

    1986

  24. [31]

    Holderrieth, P. (2021). Cores for piecewise-deterministic markov processes used in markov chain monte carlo. Electronic Communications in Probability, 26:1–12

    2021

  25. [32]

    and van der Vaart, A

    Kleijn, B. and van der Vaart, A. (2012). The Bernstein-Von-Mises theorem under misspec- ification. Electronic Journal of Statistics , 6(none):354 – 381

    2012

  26. [33]

    Krauth, W. (2021). Event-chain monte carlo: foundations, applications, and prospects. Frontiers in Physics, 9:663457

    2021

  27. [34]

    Lee, A. J. (2019). U-statistics: Theory and Practice . Routledge

    2019

  28. [35]

    and Wang, L

    Lu, J. and Wang, L. (2022). On explicit l 2-convergence rate estimate for piecewise deter- ministic markov processes in mcmc algorithms. The Annals of Applied Probability, 32(2):1333– 1361

    2022

  29. [36]

    Luczak, M. J. and Norris, J. R. (2013). Averaging over fast variables in the fluid limit for markov chains: Application to the supermarket model with memory. The Annals of Applied Probability, 23(3):957–986. 49

    2013

  30. [37]

    Neal, P., Roberts, G., and Yuen, W. K. (2012). Optimal scaling of random walk Metropolis algorithms with discontinuous target densities. The Annals of Applied Probability , 22(5):1880 – 1927. Publisher: Institute of Mathematical Statistics

    2012

  31. [38]

    and Fearnhead, P

    Nemeth, C. and Fearnhead, P. (2021). Stochastic gradient Markov chain Monte Carlo. Journal of the American Statistical Association , 116(533):433–450

    2021

  32. [39]

    Pagani, F., Chevallier, A., Power, S., House, T., and Cotter, S. (2024). Nuzz: Numerical zig-zag for general models. Statistics and Computing , 34(1):61

    2024

  33. [40]

    and Stuart, A

    Pavliotis, G. and Stuart, A. (2008). Multiscale methods: Averaging and homogenization . Springer Science & Business Media

    2008

  34. [41]

    M., and Roberts, G

    Pollock, M., Fearnhead, P., Johansen, A. M., and Roberts, G. O. (2020). Quasi-Stationary Monte Carlo and The Scale Algorithm. Journal of the Royal Statistical Society Series B: Statistical Methodology, 82(5):1167–1221

    2020

  35. [42]

    O., Gelman, A., and Gilks, W

    Roberts, G. O., Gelman, A., and Gilks, W. R. (1997). Weak convergence and optimal scaling of random walk Metropolis algorithms. The Annals of Applied Probability , 7(1):110–120

    1997

  36. [43]

    Roberts, G. O. and Rosenthal, J. S. (2001). Optimal scaling for various Metropolis-Hastings algorithms. Statistical Science, 16(4):351–367

    2001

  37. [45]

    Sahu, S. K. and Roberts, G. O. (1999). On convergence of the EM algorithmand the Gibbs sampler. Statistics and Computing , 9(1):55–64

    1999

  38. [46]

    M., Deligiannidis, G., Doucet, A., and Pitt, M

    Schmon, S. M., Deligiannidis, G., Doucet, A., and Pitt, M. K. (2021). Large-sample asymp- totics of the pseudo-marginal method. Biometrika, 108:37–51

    2021

  39. [47]

    Schmon, S. M. and Gagnon, P. (2022). Optimal scaling of random walk Metropolis algo- rithms using Bayesian large-sample asymptotics. Statistics and Computing , 32:1–16

    2022

  40. [48]

    and Roberts, G

    Vasdekis, G. and Roberts, G. O. (2021). Speed up zig-zag. arXiv preprint arXiv:2103.16620

    2021 arXiv

  41. [49]

    and Teh, Y

    Welling, M. and Teh, Y. W. (2011). Bayesian learning via stochastic gradient langevin dynamics. In Proceedings of the 28th international conference on machine learning (ICML- 11), pages 681–688. Citeseer

    2011

  42. [50]

    and Robert, C

    Wu, C. and Robert, C. P. (2020). Coordinate sampler: a non-reversible gibbs-like mcmc sampler. Statistics and Computing , 30(3):721–730

    2020

  43. [51]

    J., and Jordan, M

    Yang, Y., Wainwright, M. J., and Jordan, M. I. (2016). On the computational complexity of high-dimensional Bayesian variable selection. The Annals of Statistics , 44(6):2497–2532. Publisher: Institute of Mathematical Statistics

    2016

  44. [52]

    and Roberts, G

    Zanella, G. and Roberts, G. (2021). Multilevel Linear Models, Gibbs Samplers and Multigrid Decompositions (with Discussion). Bayesian Analysis , 16(4):1309–1391. Publisher: Interna- tional Society for Bayesian Analysis. 50

    2021

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.