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This paper shows that when a Pólya-urn quantile martingale posterior is stopped after finitely many imputations, the variance of the deployed tracker state is governed by an explicit gain factor G_a(r), equal to the familiar tail fraction o

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2026-08-01 11:33 UTC pith:ARPH4I3L

load-bearing objection The finite-horizon G_a law is real, derived-not-fitted, and the paper deserves a serious referee; the regression half is more conditional and honestly scoped.

arxiv 2607.19839 v1 pith:ARPH4I3L submitted 2026-07-22 math.ST stat.TH

Finite-horizon quantile martingale posteriors: raw-urn laws and matrix-gain regression

Nam Anh Le This is my paper
classification math.ST stat.TH MSC 62G2062G0962F1562L20
keywords martingale posteriorPólya urnquantile regressionBayesian bootstrapstochastic approximationfinite horizonBernstein–von Misesmatrix gain
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper derives exact finite-horizon laws for the empirical Pólya-urn quantile martingale posterior, the object a practitioner actually holds when simulation stops after N imputations. It proves that the stopped urn's plug-in quantile keeps the classical tail-sum variance fraction N/(n+N), but the recursively updated tracker state does not: its variance carries an additional factor G_a that can over- or under-disperse depending on the gain. When the gain is tuned to the inverse density (a=1), the factor collapses to the tail fraction, and a feasible density-free inflation restores calibration. For conditional quantile regression, the paper establishes a finite-horizon Bernstein–von Mises theorem with a full inverse-Jacobian matrix gain, and shows that scalar or diagonal gains cannot reproduce the sandwich covariance. If correct, these results supply exact finite-horizon corrections for streaming quantile trackers and quantile-regression posterior bands.

Core claim

The central claim is Theorem 1(iii): for the raw empirical Pólya-urn quantile recursion run to horizon N=⌊λn⌋, conditionally in P0-probability, √n(θ_{n+N}−q̂τ,n) | F_n ⇒ N(0, Στ G_a(r)) with r=1+λ and G_a(r) a closed-form integral. At a=cf0(qτ)=1, G_1(r)=λ/(1+λ) recovers the martingale tail fraction; at a=2, r=2 the factor exceeds one, producing overdispersion rather than the usual deficit. Corollary 1 shows the plug-in quantile of the stopped urn measure keeps the tail fraction, so the distortion is specific to the retained recursive state. In the regression setting, Theorem 3 asserts that starting a smoothed martingale posterior at the ordinary quantile-regression estimator and using a fro

What carries the argument

The load-bearing object is the factor G_a(r) = ∫_1^r {(1−a)r − a s^{a−1} − s^{−1}}^2 ds, a closed-form integral that emerges from jointly linearizing the drifted tracker update and the Pólya-urn measure martingale while retaining their shared innovations; its square-root inverse corrects the stopped state. In the regression half, the key mechanism is the frozen inverse-Jacobian matrix gain Â_n(u)=Ĵ_n(u)^{-1}, which preconditions the functional update so that the finite-horizon process covariance exactly mirrors the sandwich kernel {min(u,v)−uv} J0(u)^{-1} ΣX J0(v)^{-T}.

Load-bearing premise

The conditional-quantile regression result assumes the true model is exactly linear, Q0(u|x)=x'β0(u), with bounded regressors and a uniformly positive-definite Jacobian; if that ideal-model assumption fails, the advertised calibrated process bands are not guaranteed to hold.

What would settle it

Run recursion (1) on Uniform(0,1) data with τ=0.5, frozen gain making a=2, horizon N=n, and n large (e.g., 10^5): the scaled variance n·var(θ_{n+N}−q̂) should approach Στ G_2(2)=0.25×1.145833=0.28646, not the tail-fraction value 0.125. In the regression setting, fit the matrix-gain posterior in a design where J0 is far from scalar but set a diagonal gain: if the posterior intercept–slope correlation does not turn opposite to the target sandwich correlation, Proposition 2 is falsified.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Implementations of martingale-posterior quantiles that stop at a finite horizon must apply the G_a inflation to the retained tracker state, not the plain tail fraction N/(n+N), to achieve nominal coverage.
  • With a density-adapted gain (a→1), the correction becomes feasible and density-free: multiply the stopped increment by {(n+N)/N}^{1/2}.
  • Sharing one urn path across several quantile levels induces a joint finite-horizon law; when all gains are adapted, a common inflation restores the full joint quantile covariance, enabling calibrated simultaneous regions.
  • For conditional quantile regression, only the full inverse-Jacobian gain matches the sandwich covariance; scalar or diagonal gains misorient the posterior joint geometry, even when marginal variances are matched.
  • At infinite horizon the raw-urn posterior coincides with the Bayesian-bootstrap quantile posterior, so exact Dirichlet-weighted quantiles remain the recommended default for infinite-endpoint sampling.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The G_a law suggests a practical rule for any online quantile tracker whose state is coupled to another algorithm: periodically re-inflate the state by the estimated factor Ĝ_â(r) rather than by the tail fraction, a recipe that goes beyond the paper's theorems and could be tested on coupled streaming systems.
  • The regression result implies that diagonal-precision Bayesian or bootstrap uncertainty for conditional quantiles in positive-design problems can systematically reverse the sign of cross-quantile correlations; a cheap diagnostic is to inspect the off-diagonal of  Σ_X  against the target sandwich.
  • The atomic-support obstruction to a uniform-in-τ process version suggests a smoothed-urn analogue with shared innovations as the natural route to a full process theory; the paper leaves this as an open construction.
  • Because the matrix-gain posterior has no robustness outside the exactly linear conditional-quantile model, a cautious user should report the plain quantile-regression sandwich intervals alongside whenever misspecification is plausible.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper studies finite-horizon martingale posteriors for quantiles based on the empirical Pólya-urn predictive. For the raw stopped tracker (1) with frozen gain c, it derives the conditional limit √n(θ_{n+N}−q̂_{τ,n}) | F_n ⇒ N(0, Σ_τ G_a(1+λ)) with a = c f_0(q_τ), where G_a is given in closed form in Eq. (3). The infinite-horizon endpoint is shown to coincide with the Bayesian-bootstrap quantile, and the plug-in quantile of the stopped urn measure is shown to retain the ordinary tail-sum fraction λ/(1+λ) (Corollary 1). The paper also gives a joint law for several quantile levels driven by shared urn innovations, and a smoothed conditional-quantile-regression recursion with a full inverse-Jacobian matrix gain, proving a process Bernstein–von Mises theorem and a necessity result for the matrix gain. The finite-horizon correction is implemented with an estimated density-adapted gain, and the numerical sections include extensive simulations and an Engel-data illustration.

Significance. If correct, the paper closes a real gap: finite-horizon stopping of a quantile martingale posterior does not, in general, produce the martingale tail-sum variance, and the paper gives the exact first-order inflation factor explicitly. The G_a formula is derived from a shared-innovation linearization and a weighted-martingale representation, not fitted to the target result; the a=1 case recovers the known tail fraction, which is a strong internal consistency check. The matrix-gain regression theorem is a substantive contribution to function-valued martingale posteriors, and the necessity result clarifies why scalar or diagonal gains cannot match the sandwich covariance process. The paper is unusually transparent about scope: the exact-linearity assumption for the regression half is disclosed, the phase-boundary result is stated as conditional on Chung–Fabian limits, and the raw-urn results are explicitly limited to finitely many fixed quantile levels. The shipped code and data with reproducibility gates further strengthen confidence in the numerical claims.

minor comments (4)
  1. [Sec. 3, Eq. (3)] The display says the continuous value at a=1/2 is '(1/4) r^{-1} log r for the final term'. Since G_a itself is continuous at a=1/2, it would be clearer to state that the remaining terms are unchanged and this is the limiting value of the whole expression, or to give the full continuous formula explicitly.
  2. [Sec. 2, algorithm box] The correction step uses Ĝ_â(1+N/n) before a and G_a are formally defined in Section 3. A one-sentence forward reference or a parenthetical definition of â and r in the algorithm box would improve readability for practitioners.
  3. [Sec. 5.2, Table 2] The column 'Joint coverage' is carefully described in the text as a moment-based Hotelling-type diagnostic, but the table header alone could be mistaken for the max-standardized region of Theorem 3. A brief footnote in the table would remove this ambiguity.
  4. [Sec. 7] The paper's own statement that the raw-urn results do not extend to a uniform-in-τ process version is important and easy to miss. Consider stating this limitation at the end of Section 3 as well, so that readers do not overgeneralize the finite-collection theorem.

Circularity Check

0 steps flagged

No significant circularity: the raw-urn G_a law is derived from martingale identities and verified against external simulation, and the regression matching is an explicit construction under disclosed assumptions.

full rationale

The paper's central claim, Theorem 1(iii) with G_a(r), is derived, not fitted: Lemma S4 starts from the exact recursion identity (S10), obtains the weight kernel h_a by variation of constants and a Riemann approximation (S11), and then evaluates the resulting conditional variance as G_a(r). The proof relies on Q-local (Lemma S2), which is itself proven from the primitive density conditions rather than imported as an assumption. Corollary 1 is a separate calculation for the plug-in quantile of the stopped urn and does not presuppose the tracker law. The finite-horizon correction in Theorem 1(iii) is a standard continuous-mapping standardization of the already-derived variance, not a parameter fitted to target coverage. The matrix-gain regression result (Theorem 3) is also a construction: the inverse-Jacobian gain is placed into the update (4), and the covariance identity (S17) follows from Lemma S6-S7 under the stated primitive conditions R1-R5; the statement that this gain matches the frequentist sandwich is a mathematical consequence, not a fit of the result to itself. Proposition 1 is explicitly conditional on classical Chung-Fabian limits and even states that the subcritical limit is assumed, not newly derived, so there is no hidden citation dependence. The paper contains no self-citations by the author: the closest references (Fong and Yiu) are external prior work and are also explicitly distinguished from the new recursions. The disclosed exact-linearity assumption R1 is a scope limitation, not a circularity. No step reduces a prediction to its own input by definition or by fitted renaming.

Axiom & Free-Parameter Ledger

5 free parameters · 8 axioms · 0 invented entities

The paper introduces no new physical entities. Its load-bearing ingredients are the practitioner-chosen effective gain a, kernel density estimation defaults, and the model-specific assumptions for the regression extension. The mathematical construction (matrix-gain smoothed martingale posterior, Gaussian-copula smoothing schedule, G_a correction) is a resampling algorithm rather than a new conserved quantity or mediator.

free parameters (5)
  • effective gain a = c f0(q_tau) = not fitted; estimated by \hat a = c_n \hat f_n(\hat q)
    Indexes the variance factor G_a(r) in Theorem 1(iii)/Lemma S4. The correction requires a consistent estimate of a; if the density estimate is biased (extreme quantiles, trim binding), the corrected intervals are miscalibrated.
  • kernel bandwidth h_n = 0.9 min{s, IQR/1.34} n^{-1/5} (Silverman default)
    Used to estimate f0(q_tau) for the feasible adapted gain; theorem requires h_n→0 and n h_n / log n →∞. Sensitivity to bandwidth is reported in S4.1.
  • trim interval / scale-equivariant trim = [0.02, 1.50] dimensionless
    Keeps the inverse-density gain bounded; if the true dimensionless density lies outside this interval, the gain is capped and the correction can fail, as in the lognormal tau=0.9 cell.
  • generalized-eigenvalue floor epsilon_n = 0.28 (400/n)^{1/4}
    Regularizes Jacobian estimation in quantile regression; at Engel tau=.75, changing the floor from 0.18 to 0.38 changes the posterior covariance by 0.437/0.430, so upper-quantile scale is floor-dependent.
  • smoothing schedule parameters d_rho, k = 1, 0.8 defaults
    Defines rho_i=(1-d_rho i^{-k})^{1/2} in (4); the theorem requires 0<d_rho≤1 and 0<k<1, but finite-sample results depend on the chosen defaults.
axioms (8)
  • standard math The Pólya-urn imputed sequence is exchangeable and conditionally i.i.d. from a Dirichlet(1,...,1) mixture (de Finetti).
    Section S2, Lemma S1 uses this to define the atomic limiting target and to compute the drift; this is Blackwell-MacQueen.
  • domain assumption f0 is continuous and positive in a neighbourhood of each target quantile; the sample quantile pilot is consistent.
    Conditions Q1 and Theorem 1; needed for Q-local linearization, Bahadur inversion, and the form of Sigma_tau.
  • domain assumption Frozen gains c_n are F_n-measurable and c_n →_p c∈(0,∞).
    Condition Q2; the entire G_a parametrization is in terms of a=c f0(q_tau).
  • domain assumption Kernel density estimator consistency conditions: K bounded, Lipschitz, bounded variation, ∫|u|K(u)du<∞, h_n→0, n h_n/log n→∞, and the trim contains the true density.
    Condition Q3 / Theorem 1(iv); feasible adaptation \hat a→1 depends on this.
  • domain assumption Linear conditional quantile model Q0(u|x)=x'β0(u) with bounded design X, positive-definite Sigma_X, and Jacobian eigenvalues bounded away from 0 and infinity.
    Conditions R1-R2; Theorem 3 and Proposition 2 rely on correct specification and bounded design. The paper explicitly states there is no robustness to misspecification.
  • domain assumption Uniform quantile-regression Donsker/Bahadur representation and uniform Jacobian estimator consistency.
    Conditions R3-R4; used in Lemma S8 and Theorem 3, with primitive conditions and rate assumptions in the Supplement.
  • ad hoc to paper Classical Chung-Fabian subcritical/critical limits for the data-pass recursion are taken as assumptions in Proposition 1.
    Proposition 1 is explicitly conditional on these classical limits; the paper does not re-derive them for indicator noise.
  • ad hoc to paper The Gaussian-copula smoothing kernel H_rho and the rho_i schedule define the smoothed predictive construction.
    Section 4; this is the paper's model choice, not derived. The finite-horizon covariance formula (5) depends on it.

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read the original abstract

Martingale posteriors quantify uncertainty by forward-imputing observations from one-step-ahead predictive distributions, but implementations stop after finitely many imputations. For the empirical P\'olya-urn posterior of a quantile the law of the stopped state is derived. The quantile of the stopped urn measure keeps the familiar martingale tail-sum variance fraction; the deployed stochastic-approximation tracker with frozen gain $c$ does not. Its variance carries an explicit factor $G_a$ with $a=cf_0(q_\tau)$, which may fall below or exceed the tail fraction, and a density-adapted gain restores calibration through a density-free inflation. Shared urn innovations yield the joint law of finitely many quantile levels. For conditional quantile regression, a smoothed martingale posterior started at the ordinary quantile-regression estimator with a full inverse-Jacobian matrix gain satisfies a process Bernstein--von Mises theorem with calibrated finite-horizon bands; scalar or diagonal gains cannot match the sandwich covariance process.

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Works this paper leans on

24 extracted references · 6 canonical work pages

  1. [9]

    Retrieved 13 July 2026; SHA-256: 9af578624f649277f7ff4f3b248c274714c435601a39f8be73cc208cfb4101ea

    URL https://vincentarelbundock.github.io/Rdatasets/csv/quantreg/engel.csv. Retrieved 13 July 2026; SHA-256: 9af578624f649277f7ff4f3b248c274714c435601a39f8be73cc208cfb4101ea. Roger Koenker and Gilbert Bassett. Regression quantiles.Econometrica, 46(1):33–50,

    2026

  2. [10]

    Albert Y

    doi: 10.1080/01621459.1987.10478508. Albert Y. Lo. A large sample study of the Bayesian bootstrap.The Annals of Statistics, 15(1):360–375,

    arXiv 1987

  3. [11]

    doi: 10.1214/aos/1176350271. 34 S. P. Lyddon, C. C. Holmes, and S. G. Walker. General Bayesian updating and the loss-likelihood bootstrap. Biometrika, 106(2):465–478,

    arXiv

  4. [13]

    Ulrich K

    doi: 10.1214/23-sts914. Ulrich K. Müller. Risk of Bayesian inference in misspecified models, and the sandwich covariance matrix. Econometrica, 81(5):1805–1849,

    work page doi:10.1214/23-sts914

  5. [1954]

    doi: 10.1214/aoms/1177728716. H. A. David and H. N. Nagaraja.Order Statistics. Wiley, Hoboken, NJ, 3rd edition,

    arXiv

  6. [1958]

    Benjamin A

    doi: 10.1214/aoms/1177706619. Benjamin A. Shaby. The open-faced sandwich adjustment for MCMC using estimating functions.Journal of Computational and Graphical Statistics, 23(3):853–876,

    arXiv

  7. [1967]

    doi: 10.1214/aoms/1177699069. C.-S. Weng. On a second-order asymptotic property of the Bayesian bootstrap mean.The Annals of Statistics, 17(2):705–710,

    arXiv

  8. [1968]

    Edwin Fong and Andrew Yiu

    doi: 10.1214/aoms/1177698258. Edwin Fong and Andrew Yiu. Bayesian quantile estimation and regression with martingale posteriors. Journal of the Royal Statistical Society Series B: Statistical Methodology, page qkaf080,

    arXiv

  9. [1971]

    Donald B

    doi: 10.1016/b978-0-12-604550-5.50015-8. Donald B. Rubin. The Bayesian bootstrap.The Annals of Statistics, 9(1):130–134,

    doi:10.1016/b978-0-12-604550-5.50015-8

  10. [1973]

    Likai Chen, Georg Keilbar, and Wei Biao Wu

    doi: 10.1214/aos/1176342372. Likai Chen, Georg Keilbar, and Wei Biao Wu. Smoothed SGD for quantiles: Bahadur representation and Gaussian approximation. arXiv:2505.13299,

    arXiv

  11. [1981]

    David Ruppert

    doi: 10.1214/aos/ 1176345338. David Ruppert. Stochastic approximation. In B. K. Ghosh and P. K. Sen, editors,Handbook of Sequential Analysis, pages 503–529. Marcel Dekker, New York,

    doi:10.1214/aos/

  12. [1983]

    doi: 10.1137/0904048. A. W. van der Vaart.Asymptotic Statistics. Cambridge University Press, Cambridge,

    work page doi:10.1137/0904048

  13. [1987]

    Roger Koenker

    doi: 10.1080/07474948708836128. Roger Koenker. [dataset] engel food-expenditure data from the quantreg package. Rdatasets mirror, https://vincentarelbundock.github.io/Rdatasets/csv/quantreg/engel.csv,

    work page doi:10.1080/07474948708836128

  14. [1989]

    Yiu Yin Yung, Stephen M. S. Lee, and Edwin Fong. Moment martingale posteriors for semiparametric predictive Bayes. arXiv:2507.18148,

    Pith/arXiv arXiv

  15. [1992]

    Jens Praestgaard and Jon A

    doi: 10.1137/0330046. Jens Praestgaard and Jon A. Wellner. Exchangeably weighted bootstraps of the general empirical process. The Annals of Probability, 21(4):2053–2086,

    doi:10.1137/0330046 2053

  16. [1993]

    Mathieu Ribatet, Daniel Cooley, and Anthony C

    doi: 10.1214/aop/1176989011. Mathieu Ribatet, Daniel Cooley, and Anthony C. Davison. Bayesian inference from composite likelihoods, with an application to spatial extremes.Statistica Sinica, 22(2):813–845,

    arXiv

  17. [1996]

    doi: 10.1007/978-1-4757-2545-2. J. H. Venter. An extension of the Robbins–Monro procedure.The Annals of Mathematical Statistics, 38(1): 181–190,

    doi:10.1007/978-1-4757-2545-2

  18. [2010]

    doi: 10.3982/ECTA7880. K. L. Chung. On a stochastic approximation method.The Annals of Mathematical Statistics, 25(3):463–483,

    doi:10.3982/ecta7880

  19. [2013]

    doi: 10.3982/ecta9097. M. I. Parzen, L. J. Wei, and Z. Ying. A resampling method based on pivotal estimating functions.Biometrika, 81(2):341–350,

    work page doi:10.3982/ecta9097

  20. [2014]

    Luke Tierney

    doi: 10.1080/10618600.2013.842174. Luke Tierney. A space-efficient recursive procedure for estimating a quantile of an unknown distribution. SIAM Journal on Scientific and Statistical Computing, 4(4):706–711,

    arXiv 2013

  21. [2019]

    Blake Moya and Stephen G

    doi: 10.1093/biomet/asz006. Blake Moya and Stephen G. Walker. Martingale posterior distributions for time-series models.Statistical Science, 40(1):68–80,

    doi:10.1093/biomet/asz006

  22. [2023]

    doi: 10.1093/jrsssb/qkad005. A. Hagemann. Cluster-robust bootstrap inference in quantile regression models.Journal of the American Statistical Association, 112(517):446–456,

    doi:10.1093/jrsssb/qkad005

  23. [2025]

    Edwin Fong and Andrew Yiu

    doi: 10.1093/jrsssb/qkaf080. Edwin Fong and Andrew Yiu. Asymptotics for a class of parametric martingale posteriors.Biometrika, 113 (2):asag007,

    work page doi:10.1093/jrsssb/qkaf080

  24. [2026]

    Edwin Fong, Chris Holmes, and Stephen G

    doi: 10.1093/biomet/asag007. Edwin Fong, Chris Holmes, and Stephen G. Walker. Martingale posterior distributions.Journal of the Royal Statistical Society Series B: Statistical Methodology, 85(5):1357–1391,

    work page doi:10.1093/biomet/asag007