Pith. sign in

REVIEW 2 major objections 4 minor 4 cited by

A Computable Measure of Suboptimality for Entropy-Regularised Variational Objectives

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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

Pith's one-line read A computable score, the kernel gradient discrepancy, measures how far samples are from the target of any entropy-regularised variational objective, and driving it to zero provably characterizes convergence to that target.

desk verdict Useful new discrepancy for entropy-regularised variational objectives; main theorem's convergence control rests on an assumption that excludes standard Bayesian losses, so the 'if and only if' claim is narrower than advertised. read the letter →

arxiv 2509.10393 v4 pith:6CK7SVSF submitted 2025-09-12 stat.CO stat.ML

Clémentine Chazal , Heishiro Kanagawa , Zheyang Shen , Anna Korba , Chris. J. Oates This is my paper
classification stat.COstat.ML MSC 62F1562G0560B10
keywords gradientdiscrepancykernelSteinentropy-regularisedvariationalobjectivepost-Bayesianinferencemean-fieldLangevindynamicssamplequality
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

This paper asks a practical question: for targets defined as minimisers of an entropy-regularised objective L(Q)+KL(Q||Q0), how can we tell whether a collection of samples is close to the target when we cannot evaluate the objective on those samples? The authors introduce the gradient discrepancy, a statistical divergence that measures the size of the variational gradient of the objective at a candidate distribution, and a kernelised version, KGD, that can be computed in closed form for empirical measures. In the standard Bayesian case KGD coincides with the kernel Stein discrepancy, giving a new interpretation of that classical tool. The paper proves that under regularity conditions, KGD goes to zero if and only if the candidate distributions α-converge to the target, which justifies using KGD for tuning, comparing, and designing sampling algorithms. A sympathetic reader would recognise this as a missing diagnostic: a way to measure suboptimality for post-Bayesian and mean-field methods where unnormalised densities are unavailable.

What carries the argument

The key object is the kernel gradient discrepancy KGD_K(Q), defined as the supremum over the unit ball of a vector-valued RKHS of the integral of the operator T_Q v(x) = [∇log q0(x) − ∇V L(Q)(x)]·v(x) + ∇·v(x). This operator is the integrand obtained by integrating the variational gradient against a vector field and applying integration by parts, so it does not require the density of Q. KGD has a closed-form double-integral expression in terms of a Q-dependent kernel k^Q_K(x,x'), which reduces to the classical Stein kernel when L is a linear functional. The variational gradient ∇V J(Q) vanishes exactly at stationary points satisfying the self-consistency equation dQ/dQ0 ∝ exp(−L'(Q)), and KG

What would settle it

Take the Bayesian loss L(Q)=∫||x||² dQ(x), so ∇V L(Q)(x)=2x, with Q0 a standard Gaussian. Assumption 7(ii) fails because the variational gradient is unbounded. Construct a sequence of empirical measures Q_n with KGD(Q_n)→0 that does not α-converge to the target P ∝ exp(−||x||²). If such a sequence exists, Corollary 1 would be false without the uniform-boundedness assumption, demonstrating that the theorem's scope is exactly that assumption.

Watch

Extended reading notes

Core claim

The central claim is that the size of the variational gradient of the entropy-regularised objective J(Q)=L(Q)+KL(Q||Q0) can be turned into a computable discrepancy, the kernel gradient discrepancy (KGD), even when Q is an empirical measure and no unnormalised density for the target exists. The construction replaces the density-dependent gradient with an integration-by-parts identity that exposes a generalised score b_Q = ∇log q0 − ∇V L(Q), and KGD is the RKHS norm of the associated Stein operator. The paper's principal theoretical result is Corollary 1: under Assumptions 1–3 and 5–7, KGD(Q_n)→0 if and only if Q_n α-converges to the target P. The paper states this as: any consistent algorithm

Load-bearing premise

The central convergence characterisation relies on Assumption 7(ii): the variational gradient ∇V L(Q) must be uniformly bounded over all distributions Q and all positions x — a condition that fails for common Bayesian losses such as a Gaussian negative log-likelihood, whose score grows linearly with x.

Editorial extensions

If this is right

  • If KGD is minimised to zero by a sequence of empirical measures, those measures converge to the target in α-moment-weighted weak topology, so KGD can serve as a convergence diagnostic when the target density is inaccessible.
  • KGD provides a principled criterion for selecting the step size in mean-field Langevin dynamics, replacing the intractable variational objective with a computable proxy.
  • KGD enables the first extensible sampling algorithm for entropy-regularised variational targets, where particles are added one at a time until a desired accuracy is reached.
  • KGD permits parametric variational inference without requiring explicit densities or easy-to-compute Jacobian determinants, generalising KSD-based variational inference to nonlinear losses.
  • KGD supplies the first convergence analysis for nonlinear Stein variational gradient descent, with a non-asymptotic bound in mean-square KGD.
  • In the Bayesian linear-loss case, KGD equals KSD, so the paper provides a new variational-gradient interpretation of KSD and recovers existing KSD theory as a special case.

Reading between the lines

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

  • Beyond the paper: the equivalence between KGD convergence and α-convergence suggests that KGD could be used as a universal objective for post-Bayesian inference, with different kernel choices trading off which moments of the target are emphasised.
  • Beyond the paper: the extensible sampling construction could be combined with existing Stein thinning and MCMC-output compression methods, carrying their practical benefits to entropy-regularised objectives.
  • Beyond the paper: because KGD is a kernel method with O(d n²) cost, the paper's results imply that the substantial literature on fast KSD approximations could be transferred to KGD, potentially making the diagnostic scalable.
  • Beyond the paper: the authors' proof relies on a strong uniform boundedness of the variational gradient over all distributions; for common unbounded Bayesian losses the characterisation may fail, so practitioners should check this condition or use a compactified loss before applying Corollary 1.
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

2 major / 4 minor

Summary. The paper introduces gradient discrepancy (GD) and its kernelized instance KGD as a computable measure of suboptimality for distributions defined as minimizers of entropy-regularized variational objectives J(Q)=L(Q)+KL(Q||Q0). KGD is shown to be closed-form for empirical measures, to reduce to kernel Stein discrepancy (KSD) when L is the Bayesian negative log-likelihood, and to be usable for tuning mean-field Langevin dynamics, for extensible sampling, for parametric variational inference without explicit densities, and for a generalisation of SVGD (VGD). The theoretical core establishes that KGD characterises stationarity (Theorem 1), is continuous under α-convergence (Theorem 2), and under additional assumptions controls α-convergence (Theorem 3, Corollary 1). The paper includes detailed proofs in appendices, a recommended kernel, and code for all experiments.

Significance. If the claims hold, KGD fills a genuine gap: it provides a computable diagnostic for post-Bayesian and nonlinear-loss variational targets where unnormalised densities are unavailable. The elegant interpretation of KSD as the size of a variational gradient is a useful conceptual contribution, and the algorithmic applications (tuning MFLD, extensible sampling, VGD) are potentially impactful. The paper is technically ambitious: appendices supply careful RKHS arguments, stationarity and continuity proofs, and reproducible experimental code. However, the headline convergence-control theorem is proven only under a restrictive uniform-boundedness condition that excludes a leading motivating case, so the central claim needs qualification or repair.

major comments (2)
  1. [Assumption 5(ii)] The uniform bound sup_{Q∈P(R^d),x}∥∇_V L(Q)(x)∥<∞ and the condition L'(Q)(x)=o(‖x‖^{2γ}) are not satisfied by the standard Bayesian loss (2). For a Gaussian likelihood, ∇_V L(Q)(x)=Σ^{-1}(x−μ) is unbounded in x, and L'(Q)(x) is quadratic, not o(‖x‖^2) for γ=1. The proof of Theorem 3 in Appendix A.9 (Lemmas 14–16) uses Assumption 7(ii) to obtain uniform dissipativity of b_Q and to verify growth conditions; without it, the convergence-control argument, and hence the 'if and only if' in Corollary 1, is not established for this leading case. The reparameterization workaround in §4.3 (replace Q0 by Q̃0∝q0 exp(−u)) is only discussed for Proposition 6; if it is intended to rescue Theorem 3, the equivalence needs to be stated and proved at the level of KGD and Assumptions 5–7. This is a scope gap rather than a contradiction, and it is likely repairable (e.g., by invoking known KSD convergence-co
  2. [Assumption 5(ii)] The growth condition on ∇_i w is stated 'for each i∈{1,...,d}', but the sum over i in the variational gradient runs over the r arguments of w, not the d coordinates of R^d. This is presumably a typo (i should range over 1,...,r), but it makes the assumption formally ambiguous and should be corrected.
minor comments (4)
  1. [Section B.1.4] The main text says the reference for parametric variational inference is a standard Gaussian μ0, while the appendix says μ0=U([-3,3]^{d0}); these should be reconciled.
  2. [Equation (13)] The notation d x_t^i/dt = ∇_V KGD^2(Q_t^n)(x_t^i) is used before the functional derivative notation is fully explained for the squared KGD; a short definition or pointing to Section 2.1 would help.
  3. [Definition 6 and Appendix B] The 'recommended kernel' is used in Figures 4 and 6 with parameters α=2, β=1, c=1, but the choice of these parameters is not discussed; adding a sentence on how α and β are chosen in practice would improve usability.
  4. [General] The phrase 'first computable measure' in the Discussion should be softened to 'to our knowledge', since the paper itself notes that KSD is a computable special case for linear losses.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; KGD's stationarity characterization is by construction but convergence control is independently proved.

full rationale

KGD is introduced in Section 2.2 as the RKHS-constrained norm of the linear functional T_Q v = [(∇log q0)−∇_V L(Q)]·v + ∇·v, which, by Eq. (5), equals −∫∇_V J(Q)·v dQ. Thus KGD(Q)=0 iff the variational gradient vanishes (under universality), and since stationarity is defined by exactly that self-consistency equation (Eq. (4)), Theorem 1 is a verification of the construction rather than an independent prediction. This is not a circular derivation in the harmful sense: the paper's central new claim is convergence control (Theorem 3/Corollary 1), whose substantive 'KGD→0 ⇒ Q_n→P' direction is obtained through continuity (Theorem 2), separability, and a uniform-integrability argument (Lemma 13) that invokes an external lemma from Kanagawa et al. [2025] with assumptions verified in Lemma 16; the cited lemma is about KSD and does not assume the target result, and co-authorship of the cited work does not make the appeal circular because the assumptions are independently checked in the present paper. Assumption 7(ii) (uniform boundedness of ∇_V L over all Q and x) is restrictive and can fail for Bayesian losses with unbounded score, but this is a scope/validity limitation, not a circularity. The experimental self-assessment via KGD is acknowledged in Remark 5 and does not enter the derivation chain.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

The central construction introduces no new physical or conceptual entities; it defines a functional (KGD) in terms of quantities already present in the problem (nabla_V L, q0, kernel). The axioms are regularity and structural assumptions, the strongest being the uniform boundedness of the variational gradient in Assumption 7(ii). The kernel hyperparameters (alpha, beta, c, bandwidths) are free choices of the method.

free parameters (1)
  • Kernel K and its hyperparameters (e.g., alpha, beta, c in Definition 6; bandwidths in experiments) = alpha=2, beta=1, c=1 in Figures 4 and 6; inverse multi-quadric bandwidths 0.1, 4, 0.03, 0.01, 0.001, 0.1 used in experim
    KGD is defined with respect to a user-chosen kernel. Its numerical values, its continuity, and its convergence-control properties depend on this choice. The recommended kernel in Definition 6 requires the user to set alpha, beta, c to match the moment growth of the loss, so they are free choices of the method rather than fitted constants in a prediction.
assumptions (7)
  • domain assumption The loss L admits a variational gradient nabla_V L(Q) for each Q in P(R^d) (Assumption 1)
    Used throughout to define GD in terms of nabla_V L(Q); without it the gradient discrepancy is undefined. Section 2.2 and Definition 3.
  • domain assumption q0 has a positive density (Assumption 2)
    Required for the density q0 to appear in T_Q and for the integration-by-parts derivations. Stated in Section 2.2.
  • domain assumption q0 is C^1 and nabla_V L(Q) is C^0 for each Q (Assumption 3)
    Needed for the stationarity characterisation Theorem 1, where the functional D_Q is required to lie in C^1-space of vector fields.
  • domain assumption Growth and universality conditions of Assumption 4, including theta-bounded scores and a tilted universal kernel K(x,y)=L(x,y)/(theta(x)theta(y))
    Needed for Theorem 1 and Proposition 4 to ensure KGD=0 implies stationarity by separating non-stationary distributions from zero in the RKHS.
  • domain assumption Loss is of interaction energy form with controlled growth (Assumption 5)
    Needed for Theorem 2 continuity: only for such losses can KGD be rewritten as a Q-independent integral against Q^(ell+2), which is what makes alpha-continuity work. Acknowledged in Section 4.2 as covering the paper's examples.
  • domain assumption Unique stationary point P of J (Assumption 6)
    Required to speak of 'the' target P when characterising convergence; Proposition 6 gives convexity-based sufficient conditions.
  • domain assumption Generalised dissipativity plus sup_Q,x ||nabla_V L(Q)(x)|| < inf (Assumption 7)
    Assumption 7(i) requires dissipativity of nabla log q0 and beta bounds; Assumption 7(ii) requires the variational gradient to be uniformly bounded for all Q and x. The boundedness is very strong and is load-bearing for the convergence-control proof (Theorem 3 and Lemma 14). It rules out standard Bayesian losses with unbounded score.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A Computable Measure of Suboptimality for Entropy-Regularised Variational Objectives." pith.science (2026). https://pith.science/paper/6CK7SVSF

@misc{pith2026250910393,
  author       = {Pith},
  title        = {Pith review of: A Computable Measure of Suboptimality for Entropy-Regularised Variational Objectives},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6CK7SVSF}},
  note         = {Machine review of arXiv:2509.10393}
}
read the original abstract

Several methods in statistics and machine learning target a probability distribution for which an entropy-regularised variational objective is minimised. This increased flexibility introduces a computational challenge, as one loses access to an explicit unnormalised density for the target. To mitigate this difficulty, we introduce a novel measure of suboptimality called 'gradient discrepancy', and in particular a kernel gradient discrepancy (KGD) that can be explicitly computed. In the Bayesian statistics context, KGD coincides with the kernel Stein discrepancy (KSD), and we obtain a novel characterisation of KSD as measuring the size of a variational gradient. Outside this familiar setting, KGD enables novel sampling algorithms to be developed and compared, even when unnormalised densities cannot be obtained. To illustrate this point several novel algorithms are proposed and studied, including a natural generalisation of Stein variational gradient descent, with applications to mean-field neural networks and predictively oriented posteriors presented. On the theoretical side, our principal contribution is to establish sufficient conditions for desirable properties of KGD, such as continuity and convergence control.

Figures

Figures reproduced from arXiv: 2509.10393 by the authors.

Figure 1
Figure 1. KGD provides a computable measure of sample quality for approximations [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Comparing mean field Langevin dynamics (MFLD) (left) with new sampling [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Comparing mean field Langevin dynamics (MFLD) (left) with two new sampling [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Reproduction of Figure [PITH_FULL_IMAGE:figures/full_fig_p065_4.png]
Figure 5
Figure 5. Figure 5: Examining the approximations produced by MFLD, extensible sampling, and VGD [PITH_FULL_IMAGE:figures/full_fig_p068_5.png]
Figure 6
Figure 6. Figure 6: Reproduction of Figure [PITH_FULL_IMAGE:figures/full_fig_p069_6.png]

Discussion (0). Sign in to comment.

Forward citations

Cited by 4 Pith papers

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

  1. Thinned Mean Field Langevin Dynamics

    cs.LG 2026-05 unverdicted novelty 7.0 of 10

    KT-MFLD thins the particle system in mean-field Langevin dynamics to O(N^{3/2}) complexity with convergence guarantees matching standard MFLD up to logarithmic factors.

  2. Predictively-Oriented Kalman Filtering

    stat.ME 2026-06 unverdicted novelty 6.0 of 10

    EKF-PrO is a fast linear-Gaussian approximation to predictively-oriented posteriors for online filtering, with no tunable hyperparameters and cost comparable to standard methods.

  3. Concentration and Calibration in Predictive Bayesian Inference

    stat.ME 2026-05 unverdicted novelty 6.0 of 10

    Predictive Bayesian inference posteriors concentrate onto a forward-model-dependent quantity and produce miscalibrated credible sets unless the predictive model contains the true data-generating process.

  4. Predictively Oriented Posteriors

    stat.ME 2025-10 conditional novelty 6.0 of 10

    A new posterior family scores the posterior predictive directly, giving slower concentration but better predictive performance under model misspecification.

Reference graph

Works this paper leans on

16 extracted references · 6 linked inside Pith · cited by 4 Pith papers

  1. [2]

    Thus the assumptions of Kanagawa et al

    + |1−τ| 2 ∥x∥ c2 +∥x∥ 2 2 c2 +∥x∥ 2 2 −τ /2 q c2 +∥x∥ 2 2 ≤ C+c −τ CA τ + |1−τ| 4c q c2 +∥x∥ 2 2 and ∥m(x)∥op =|a(x)| ≤(1∨(c 2 + 1)1−τ ) q c2 +∥x∥ 2 2. Thus the assumptions of Kanagawa et al. [2025] are satisfied. 63 B Experimental Detail This appendix contains full experimental details for reproducing Example 2 (Section B.1) and Example 3 (Section B.2) i...

    2025

  2. [5]

    Korba, P.-C

    24 A. Korba, P.-C. Aubin-Frankowski, S. Majewski, and P. Ablin. Kernel stein discrepancy descent. InInternational Conference on Machine Learning, pages 5719–5730. PMLR,

  3. [7]

    PMLR, 2022. 24 I. Lytras and P. Mertikopoulos. Tamed Langevin sampling under weaker conditions.arXiv preprint arXiv:2405.17693, 2024. 22 S. Lyu. Interpretation and generalization of score matching. InProceedings of the Twenty- Fifth Conference on Uncertainty in Artificial Intelligence, pages 359–366, 2009. 23 29 A. Masegosa. Learning under model misspecif...

    arXiv 2022

  4. [11]

    Simon-Gabriel, A

    24 C.-J. Simon-Gabriel, A. Barp, B. Sch¨ olkopf, and L. Mackey. Metrizing weak convergence with maximum mean discrepancies.Journal of Machine Learning Research, 24(1), 2023. 50 A. Smola, A. Gretton, L. Song, and B. Sch¨ olkopf. A Hilbert space embedding for distribu- tions. InInternational Conference on Algorithmic Learning Theory, pages 13–31. Springer,

    2023

  5. [14]

    The linear growth conditions can be verified as follows

    As a result, the desired claim holds withα being replaced withaα, wherea >0 is a constant such thata∥·∥ 2 ≤ ∥·∥. The linear growth conditions can be verified as follows. LetA >0 be a constant such that∥·∥ ≤A∥·∥2. By assumption, there existsQ-independentC >0 such that∥b Q(x)∥ ≤ C(1 +∥x∥ τ ), and thus 2∥b(x)∥ ≤ |a1−τ (x)|(∥b Q(x)∥+∥∇loga 1−τ (x)∥) ≤ C(1 +A τ ∥x∥τ

  6. [16]

    The data-generating parameters wereα= logit −1(−1), β= logit −1(−3),γ= 0.4,δ= 0.02,ξ 1 = 10,ξ 2 = 15,σ= 1

    all parameters are fixed and known except forx 1 := logit(α) andx 2 = logit(β/α), which are to be inferred (i.e.d= 2). The data-generating parameters wereα= logit −1(−1), β= logit −1(−3),γ= 0.4,δ= 0.02,ξ 1 = 10,ξ 2 = 15,σ= 1. Data were simulated at times ti ranging over a uniform grid from 0 ton= 60 with increments of 1. B.2.2 Predictively Oriented Poster...

    2025

  7. [1967]

    ISBN 9780080873374. 46 C. Wang, Y. Chen, H. Kanagawa, and C. J. Oates. Steinπ-importance sampling.Advances in Neural Information Processing Systems, 36:71948–71994, 2023. 24 D. Wang and Q. Liu. Nonlinear Stein variational gradient descent for learning diversified mixture models. InInternational Conference on Machine Learning, pages 6576–6585. PMLR, 2019. ...

    arXiv 2023

  8. [1996]

    Alquier et al

    5 P. Alquier et al. User-friendly introduction to PAC-Bayes bounds.Foundations and Trends in Machine Learning, 17(2):174–303, 2024. 2 L. Ambrosio, N. Gigli, and G. Savar´ e.Gradient Flows: In Metric Spaces and in the Space of Probability Measures. Springer Science & Business Media, 2008. 41 A. Anastasiou, A. Barp, F.-X. Briol, B. Ebner, R. E. Gaunt, F. Gh...

    2024

Show all 16 references
  1. [2007]

    2, 8, 13 B. K. Sriperumbudur, K. Fukumizu, and G. R. Lanckriet. Universality, characteristic kernels and RKHS embedding of measures.Journal of Machine Learning Research, 12(7), 2011. 8 C. Stein. A bound for the error in the normal approximation to the distribution of a sum of ...

    2011

  2. [2009]

    Shen and Z

    56 Z. Shen and Z. Wang. Entropy-dissipation informed neural network for McKean–Vlasov type PDEs.Advances in Neural Information Processing Systems, 36:59227–59238, 2023. 24 Z. Shen, J. Knoblauch, S. Power, and C. J. Oates. Prediction-centric uncertainty quantifica- tion via MMD...

    2023

  3. [2016]

    Gorham and L

    13 J. Gorham and L. Mackey. Measuring sample quality with Stein’s method.Advances in Neural Information Processing Systems, 28, 2015. 3, 7 J. Gorham and L. Mackey. Measuring sample quality with kernels. InInternational Confer- ence on Machine Learning, pages 1292–1301. PMLR, 2...

    2015 arXiv

  4. [2017]

    8, 45 F. Otto. The geometry of dissipative evolution equations: The porous medium equation. Communications in Partial Differential Equations, 26(1-2):101–174, 2001. 5 G. Papamakarios, E. Nalisnick, D. J. Rezende, S. Mohamed, and B. Lakshminarayanan. Normalizing flows for proba...

    2001

  5. [2021]

    Lai and Y

    12, 24 J. Lai and Y. Yao. Predictive variational inference: Learn the predictively optimal posterior distribution.arXiv preprint arXiv:2410.14843, 2024. 3, 13 N. Lanzetti, S. Bolognani, and F. D¨ orfler. First-order conditions for optimization in the Wasserstein space.SIAM Jou...

    2024 arXiv

  6. [2022]

    Nitanda, A

    9 A. Nitanda, A. Lee, D. T. X. Kai, M. Sakaguchi, and T. Suzuki. Propagation of chaos for mean-field Langevin dynamics and its application to model ensemble.arXiv preprint arXiv:2502.05784, 2025. 9 C. J. Oates, M. Girolami, and N. Chopin. Control functionals for Monte Carlo in...

    2025 arXiv

  7. [2024]

    Bauer.Measure and Integration Theory

    3, 18, 22, 44, 46, 48, 49, 50, 53, 54, 61 H. Bauer.Measure and Integration Theory. De Gruyter, 2001. ISBN 9783110866209. 46 C. B´ enard, B. Staber, and S. Da Veiga. Kernel Stein discrepancy thinning: A theoretical perspective of pathologies and a practical fix with regularizat...

    2001

  8. [2025]

    22 26 M. Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport.Advances in Neural Information Processing Systems, 26, 2013. 24 P. Del Moral. Mean field simulation for Monte Carlo integration.Monographs on Statistics and Applied Probability, 126(26):6, 2013. 3...

    2013 arXiv

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.