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arxiv: 2606.21707 · v1 · pith:EMFWRIBLnew · submitted 2026-06-19 · 📊 stat.ME

Quasi-Bayes empirical Bayes estimation of sums of random variables

Stefano Favaro , Sandra Fortini This is my paper

Pith reviewed 2026-06-26 13:18 UTC · model grok-4.3

classification 📊 stat.ME
keywords quasi-Bayesempirical Bayesmixture modelsNewton's algorithmplug-in estimationasymptotic consistencycredible intervalsnonparametric methods
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The pith

Quasi-Bayes empirical Bayes uses Newton's algorithm to estimate sums of random variables under mixture models.

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

The paper develops a nonparametric quasi-Bayes empirical Bayes method to estimate sums of functions involving both observed and unobserved variables in mixture models. Existing methods often require parametric assumptions or are limited in scope, so the new approach uses recursive estimation of the mixing distribution with Newton's algorithm. This produces a plug-in estimate that works for many utility functions, is computationally efficient, and includes asymptotic credible intervals from a central limit theorem. Theoretical results show the estimates merge with full Bayes estimates for large samples and are consistent when the model is correctly specified. Data analyses confirm it performs well compared to other empirical Bayes techniques.

Core claim

The quasi-Bayes empirical Bayes methodology addresses limitations through recursive estimation of the mixing distribution based on Newton's algorithm, yielding a computationally efficient plug-in estimate applicable to a broad class of utility functions with asymptotic credible intervals, and establishes large sample guarantees via merging with Bayes estimates and consistency under a correctly specified frequentist model.

What carries the argument

Recursive estimation of the mixing distribution based on Newton's algorithm, which produces the plug-in estimate for the target sum.

If this is right

  • The method yields computationally efficient and scalable plug-in estimates for the target sums.
  • It applies to a broad class of utility functions beyond limited nonparametric cases.
  • Asymptotic credible intervals follow from a Gaussian central limit theorem.
  • Quasi-Bayes estimates merge with Bayes estimates in large samples.
  • Consistency holds under a correctly specified frequentist model.

Where Pith is reading between the lines

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

  • The recursive updates could support online estimation in streaming data applications.
  • The approach might extend to other latent variable problems in mixture models such as prediction tasks.
  • Trade-offs between this method and fully nonparametric Bayesian alternatives could be examined in terms of speed and accuracy.
  • The asymptotic merging property suggests possible use in settings where full Bayes computation is prohibitive.

Load-bearing premise

The frequentist model is correctly specified for the consistency guarantees to hold.

What would settle it

A simulation where data comes from a misspecified mixture model and the quasi-Bayes estimates do not converge to the true sum values as sample size increases would disprove the consistency result.

Figures

Figures reproduced from arXiv: 2606.21707 by Sandra Fortini, Stefano Favaro.

Figure 1
Figure 1. Figure 1: Weibull prior, S1,n. Left panel: true values n−1S1,n (Grey o-) and estimates n−1Sˆ [O] r,n (Black .-); n−1Sˆ [ML] 1,n (Blue .-), n−1Sˆ [B] 1,n (Cyan .-), n−1Sˆ [“u,v”] 1,n (Green .-) and n−1Sˆ [Q-B] 1,n (Red .-). Right panel: MAD of Sˆ [O] r,n (Black .-); Sˆ [ML] 1,n (Blue .-), Sˆ [B] 1,n (Cyan .-), Sˆ [“u,v”] 1,n (Green .-) and Sˆ [Q-B] 1,n (Red .-) [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Weibull prior, S3,n. Left panel: true values of n−1S3,n (Grey o-) and estimates n−1Sˆ [O] 3,n (Black .-); n−1Sˆ [ML] 3,n (Blue .-), n−1Sˆ [B] 3,n (Cyan .-) and n−1Sˆ [Q-B] 3,n (Red .-). Right panel: MAD of Sˆ [O] 3,n (Black .-); Sˆ [ML] 3,n (Blue .-), Sˆ [B] 3,n (Cyan .-) and Sˆ [Q-B] 3,n (Red .-) Bayes EB method is the most attractive compromise among the parametric and nonparametric methods considered. A… view at source ↗
Figure 3
Figure 3. Figure 3: Left panel: true values T1,n(κ) (Grey o-) and estimates Tˆ [ML] 1,n (κ) (Blue .-), Tˆ [B] 1,n(κ) (Cyan .-), Tˆ [“u,v”] 1,n (κ) (Green .-) and Tˆ [Q-B] 1,n (κ) (Red .-). Right panel: AD of Tˆ [ML] 1,n (κ) (Blue .-), Tˆ [B] 1,n(κ) (Cyan .-), Tˆ [“u,v”] 1,n (κ) (Green .-) and Tˆ [Q-B] 1,n (κ) (Red .-) As κ increases, [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Left panels with centers (C, first row), left wings (LW, second row), right wings (RW, third row) and defenseman (D, fourth row): true values T1,n(κ) (Grey o-) and estimates Tˆ [ML] 1,n (κ) (Blue .-), Tˆ [B] 1,n(κ) (Cyan .-), Tˆ [“u,v”] 1,n (κ) (Green .-) and Tˆ [Q-B] 1,n (κ) (Red .-). Right panels with centers (C, first row), left wings (LW, second row), right wings (RW, third row) and defenseman (D, four… view at source ↗
Figure 5
Figure 5. Figure 5: and [PITH_FULL_IMAGE:figures/full_fig_p033_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Uniform prior, S3,n. Left panel: true values of n−1S3,n (Grey o-) and estimates n−1Sˆ [O] 3,n (Black .-), n−1Sˆ [ML] 3,n (Blue .-), n−1Sˆ [B] 3,n (Cyan .-) and n−1Sˆ [Q-B] 3,n (Red .-). Right panel: MAD of Sˆ [O] 3,n (Black .-), Sˆ [ML] 3,n (Blue .-), Sˆ [B] 3,n (Cyan .-) and Sˆ [Q-B] 3,n (Red .-) [PITH_FULL_IMAGE:figures/full_fig_p034_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Uniform prior, S1,n: oracle credible intervals (black) and quasi-Bayes credible intervals (red) CPU time refers to the time (in seconds) for processing a new observation on a laptop MacBook Pro (M1 type processor) [PITH_FULL_IMAGE:figures/full_fig_p034_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Uniform prior, S3,n: oracle credible intervals (black) and quasi-Bayes credible intervals (red) over Θ; iii) the learning rate αn = (1 + n) −0.99. This is precisely the initialization considered in the second column of [PITH_FULL_IMAGE:figures/full_fig_p035_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Weibull prior, S1,n: oracle credible intervals (black) and quasi-Bayes credible intervals (red) [PITH_FULL_IMAGE:figures/full_fig_p036_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Weibull prior, S3,n: oracle credible intervals (black) and quasi-Bayes credible intervals (red) [PITH_FULL_IMAGE:figures/full_fig_p036_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Half-Gaussian prior, S1,n. Left panel: true values n−1S1,n (Grey o-) and estimates n−1Sˆ [O] 1,n (Black .-), n−1Sˆ [ML] 1,n (Blue .-), n−1Sˆ [B] 1,n (Cyan .-), n−1Sˆ [“u,v”] 1,n (Green .-) and n−1Sˆ [Q-B] 1,n (Red .-). Right panel: MAD of Sˆ [O] 1,n (Black .-), Sˆ [ML] 1,n (Blue .-), Sˆ [B] 1,n (Cyan .-), Sˆ [“u,v”] 1,n (Green .-) and Sˆ [Q-B] 1,n (Red .-) [PITH_FULL_IMAGE:figures/full_fig_p037_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: and [PITH_FULL_IMAGE:figures/full_fig_p037_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Half-Gaussian prior, S1,n: oracle credible intervals (black) and quasi-Bayes credible intervals (red) [PITH_FULL_IMAGE:figures/full_fig_p038_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Half-Gaussian prior, S3,n: oracle credible intervals (black) and quasi-Bayes credible intervals (red) D.1.4 Square-root of half-Cauchy prior For i = 1, . . . , 100, let Xi = X1:100i denote a dataset of size n = 100i generated from the Poisson mixture model (21), with a square-root of half-Cauchy prior G, namely the distribution of the square-root of the positive part of a standard Cauchy random variable. … view at source ↗
Figure 15
Figure 15. Figure 15: and [PITH_FULL_IMAGE:figures/full_fig_p039_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Square-root of half-Cauchy prior, S3,n. Left panel: true values of n−1S3,n (Grey o-) and estimates n−1Sˆ [O] 3,n (Black .-), n−1Sˆ [ML] 3,n (Blue .-), n−1Sˆ [B] 3,n (Cyan .-) and n−1Sˆ [Q-B] 3,n (Red .-). Right panel: MAD of Sˆ [O] 3,n (Black .-), Sˆ [ML] 3,n (Blue .-), Sˆ [B] 3,n (Cyan .-) and Sˆ [Q-B] 3,n (Red .-) [PITH_FULL_IMAGE:figures/full_fig_p040_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Square-root of half-Cauchy prior, S1,n: oracle credible intervals (black) and quasi-Bayes credible intervals (red) E Additional real-data experiments E.1 European automobile insurance data We apply the quasi-Bayes EB approach to the benchmark automobile insurance claims dataset (Efron and Hastie, 2021, [PITH_FULL_IMAGE:figures/full_fig_p040_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Square-root of half-Cauchy prior, S3,n: oracle credible intervals (black) and quasi-Bayes credible intervals (red) automobile insurance company on n = 9, 461 policyholders. From the first two rows of [PITH_FULL_IMAGE:figures/full_fig_p041_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Gaussian prior, S1,n. Left panel: true values of n−1S1,n (Grey o-) and estimates n−1Sˆ [O] 1,n (Black .-), n−1Sˆ [ML] 1,n (Blue .-), n−1Sˆ [B] 1,n (Cyan .-) and n−1Sˆ [Q-B] 1,n (Red .-). Right panel: MAD of Sˆ [O] 1,n (Black .-), Sˆ [ML] 1,n (Blue .-), Sˆ [B] 1,n (Cyan .-) and Sˆ [Q-B] 1,n (Red .-) [PITH_FULL_IMAGE:figures/full_fig_p044_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Gaussian prior, S3,n. Left panel: true values of n−1S3,n (Grey o-) and estimates n−1Sˆ [O] 1,n (Black .-), n−1Sˆ [ML] 1,n (Blue .-), n−1Sˆ [B] 3,n (Cyan .-), n−1Sˆ [“u,v”] 3,n (Green .-) and n−1Sˆ [Q-B] 3,n (red .-). Right panel: MAD of Sˆ [O] 1,n (Black .-), Sˆ [ML] 3,n (Blue .-), Sˆ [B] 3,n (Cyan .-), Sˆ [“u,v”] 3,n (Green .-) and Sˆ [Q-B] 3,n (red .-) not available in this case. With regards to the est… view at source ↗
Figure 21
Figure 21. Figure 21: Gaussian prior, S1,n: oracle credible intervals (black) and quasi-Bayes credible intervals (red) G Multidimensional extension The proposed quasi-Bayes EB methodology extends naturally, with only minor modifications, to multidimensional settings under a coordinate-wise independence assumption. Let Xi = (Xi,1, . . . , Xi,d) be a d-dimensional random vector with independent coordinates, i = 1, . . . , n, whe… view at source ↗
Figure 22
Figure 22. Figure 22: Gaussian prior, S3,n: oracle credible intervals (black) and quasi-Bayes credible intervals (red) Given a measurable vector-valued utility function u : X d × Θd → R s , we consider the estimation of Sn = Xn i=1 u(Xi , θi). In this multidimensional framework, Newton’s algorithm extends verbatim, yielding a recursive update of the mixing distribution Gn on Θd . Specifically, the Newton’s algorithm becomes Gn… view at source ↗
Figure 23
Figure 23. Figure 23: Weibull prior, S1,n. Left panel: true values n−1S1,n (Grey o-) and estimates n−1Sˆ [O] 1,n (Black .-), n−1Sˆ [ML] 1,n (Blue .-), n−1Sˆ [B] 1,n (Cyan .-), Sˆ [N-ML] 1,n (Magenta .-) and n−1ˆ1 [Q-B] 1,n (Red .-). Right panel: MAD of Sˆ [O] 1,n (Black .-), Sˆ [ML] 1,n (Blue .-), Sˆ [B] 1,n (Cyan .-), Sˆ [N-ML] 1,n (Magenta .-) and Sˆ [Q-B] 1,n (Red .-) [PITH_FULL_IMAGE:figures/full_fig_p048_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Weibull prior, S3,n. Left panel: true values n−1S3,n (Grey o-) and estimates n−1Sˆ [O] 3,n (Black .-), n−1Sˆ [ML] 3,n (Blue .-), n−1Sˆ [B] 3,n (Cyan .-), Sˆ [N-ML] 3,n (Magenta .-) and n−1Sˆ [Q-B] 3,n (Red .-). Right panel: MAD of Sˆ [O] 3,n (Black .-), Sˆ [ML] 3,n (Blue .-), Sˆ [B] 3,n (Cyan .-), Sˆ [N-ML] 3,n (Magenta .-) and Sˆ [Q-B] 3,n (Red .-) population, the probability of discovering a new species… view at source ↗
read the original abstract

The estimation of sums of functions of observable and unobservable variables is a long-standing problem in statistics with applications across many domains. Empirical Bayes methods provide a natural framework for this task under mixture models, but existing approaches often rely on restrictive parametric assumptions or apply only to limited classes of functionals in nonparametric settings. We propose a nonparametric methodology, referred to as quasi-Bayes empirical Bayes, that addresses these limitations through a recursive estimation of the mixing distribution based on Newton's algorithm. The resulting plug-in estimate of the target sum is computationally efficient, scalable, and applicable to a broad class of utility functions, while enabling uncertainty quantification via asymptotic credible intervals derived from a Gaussian central limit theorem. We establish large sample asymptotic theoretical guarantees by proving a merging between the quasi-Bayes and Bayes estimates and by showing consistency under a correctly specified frequentist model. Synthetic-data and real-data analyses demonstrate the practical accuracy and stability of the method, with performance comparable to, and in some cases better than, existing empirical Bayes procedures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proposes a quasi-Bayes empirical Bayes methodology for estimating sums of functions of observable and unobservable variables under mixture models. It employs recursive estimation of the mixing distribution via Newton's algorithm to obtain a computationally efficient plug-in estimator applicable to a broad class of utility functions. The approach supplies asymptotic credible intervals derived from a Gaussian central limit theorem and establishes large-sample guarantees via a merging result between the quasi-Bayes and Bayes estimates together with consistency under a correctly specified frequentist model. Performance is illustrated through synthetic-data and real-data experiments.

Significance. If the stated asymptotic results hold, the contribution supplies a scalable nonparametric procedure for a practically relevant class of functionals that avoids restrictive parametric assumptions while furnishing built-in uncertainty quantification. The merging property with Bayes estimates and the explicit consistency statement under correct specification would constitute substantive theoretical advances in empirical Bayes methodology for sums involving latent variables.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'broad class of utility functions' is repeated without a precise characterization; a short sentence listing the functional forms covered (e.g., indicators, linear, or bounded continuous) would clarify the scope.
  2. The description of Newton's algorithm for recursive mixing-distribution estimation would benefit from an explicit statement of the update rule and the stopping criterion used in the implementation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The referee's description accurately reflects the paper's contributions on quasi-Bayes empirical Bayes estimation for sums under mixture models, including the recursive mixing distribution estimation, asymptotic guarantees, and uncertainty quantification.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper proposes a nonparametric quasi-Bayes empirical Bayes estimator via Newton's algorithm recursion on the mixing distribution, with plug-in estimates for sums of functionals and asymptotic credible intervals from a Gaussian CLT. Large-sample guarantees are established by proving merging with Bayes estimates plus consistency under a correctly specified frequentist model. These are standard asymptotic arguments that do not reduce by construction to fitted parameters, self-definitions, or self-citation chains. No load-bearing step in the abstract or described claims exhibits any of the enumerated circularity patterns; the derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no specific details on free parameters, axioms, or invented entities; none can be identified from the given text.

pith-pipeline@v0.9.1-grok · 5695 in / 1113 out tokens · 20994 ms · 2026-06-26T13:18:59.426771+00:00 · methodology

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Works this paper leans on

46 extracted references · 5 canonical work pages · 2 internal anchors

  1. [1]

    and Cappello, L

    Battiston, M. and Cappello, L. (2025). New (and old) predictive schemes with a.c.i.d. sequences. Preprint arXiv:2507.21874

    work page arXiv 2025

  2. [2]

    and Pannekoek, J

    Bethlehem, J.G., Keller, W.J. and Pannekoek, J. (1990). Disclosure control of microdata. J. Am. Statist. Assoc. 85 38--45

    1990

  3. [3]

    Bissiri, P.G., Holmes, C.C., and Walker, S.G. (2007). A general framework for updating belief distributions. J. R. Statist. Soc. B 78, 1103--1130

    2007

  4. [4]

    and Ritov, Y

    Brown, L.D., Greenshtein, E. and Ritov, Y. (2013). The Poisson compound decision problem revisited. J. Am. Statist. Assoc. 108, 741--749

    2013

  5. [5]

    and Fitzpatrick, M

    Bunge, J. and Fitzpatrick, M. (1993) Estimating the number of species: a review. J. Am. Statist. Assoc. 88, 364-373

    1993

  6. [6]

    Universal priors: solving empirical Bayes via Bayesian inference and pretraining

    Cannella, N., Teh, A., Han, Y. and Polyanskiy, Y. (2026) Universal priors: solving empirical Bayes via Bayesian inference and pretraining. Preprint arXiv:2602.15136

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  7. [7]

    and Lindley, D.V

    Deely, J.J. and Lindley, D.V. (1981). Bayes empirical Bayes. J. Am. Statist. Assoc. 76, 833--841

    1981

  8. [8]

    Efron, B. (2014). Two modeling strategies for empirical Bayes estimation. Statist. Sci. 29, 285--301

    2014

  9. [9]

    Efron, B. (2019). Bayes, oracle Bayes and empirical Bayes. Statist. Sci. 34, 177--201

    2019

  10. [10]

    Efron. B. and Hastie, T. (2021). Computer age statistical inference: algorithms, evidence, and data science. Cambridge University Press

    2021

  11. [11]

    and Thisted, R

    Efron, B. and Thisted, R. (1976). Estimating the number of unseen species: How many words did Shakespeare know? Biometrika 63, 435--447

    1976

  12. [12]

    Quasi-Bayes empirical Bayes: a sequential approach to the Poisson compound decision problem

    Favaro, S. and Fortini, S. (2024). Quasi-Bayes empirical Bayes: a sequential approach to the Poisson compound decision problem. Preprint arXiv:2411.07651

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  13. [13]

    and Teh, Y.W

    Favaro, S. and Teh, Y.W. (2013). MCMC for normalized random measure mixture models. Statist. Sci. 28, 335--359

    2013

  14. [14]

    Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1, 209--230

    1973

  15. [15]

    and Walker, S

    Fong, E., Holmes, C. and Walker, S. G. (2023). Martingale posterior distributions. J. R. Statist. Soc. B 85, 1357--1391

    2023

  16. [16]

    and Petrone, S

    Fortini, S. and Petrone, S. (2020). Quasi-Bayesian properties of a procedure for sequential learning in mixture models. J. R. Statist. Soc. B 82, 1087--1114

    2020

  17. [17]

    and Petrone, S

    Fortini, S. and Petrone, S. (2025). Exchangeability, Prediction and Predictive Modeling in Bayesian Statistics. Statist. Sci. 40, 40--67

    2025

  18. [18]

    Good, I.J. (1953). The population frequencies of species and the estimation of population parameters. Biometrika 40, 237-264

    1953

  19. [19]

    and Toulmin, G.H

    Good, I.J. and Toulmin, G.H. (1956). The number of new species, and the increase in population coverage, when a sample is increased. Biometrika 43, 45--63

    1956

  20. [20]

    and Walker, S.G

    Hahn, P.R., Martin, R. and Walker, S.G. (2018). On recursive Bayesian predictive distributions. J. Am. Statist. Assoc. 113, 1085--1093

    2018

  21. [21]

    and Kankanala, S

    Ignatiadis, N. and Kankanala, S. (2026). Compound decisions and empirical Bayes via Bayesian nonparametrics. Preprint arXiv:2602.20115

    work page arXiv 2026

  22. [22]

    and Wu, Y

    Jana, S., Polyanskiy, Y. and Wu, Y. (2025). Optimal empirical Bayes estimation for the Poisson model via minimum-distance methods. Inf. Inference 14, 1--42

    2025

  23. [23]

    Knoblauch, J., Jewson, J., and Damoulas, T. (2022). An optimization-centric view on Bayes’ rule: reviewing and generalizing variational inference. J. Mach. Learn. Res. 23, 1--109

    2022

  24. [24]

    Lindsay, B.G. (1995). Mixture models: theory, geometry and applications. NSF-CBMS Regional Conference Series in Probability and Statistics

    1995

  25. [25]

    Lo, A.Y. (1984). On a class of Bayesian nonparametric estimates. I. Density estimates Ann. Statist. 12, 351--357

    1984

  26. [26]

    and Lindsay, B.G

    Mao, C.X. and Lindsay, B.G. (2004). A Poisson model for the coverage problem with a genomic application. Biometrika 89, 669--682

    2004

  27. [27]

    Martin, R. (2012). Convergence rate for predictive recursion estimation of finite mixtures. Stat. Probab. Lett. 82, 378--384

    2012

  28. [28]

    and Ghosh, J.K

    Martin, R. and Ghosh, J.K. (2008). Stochastic approximation and Newton’s estimate of a mixing distribution. Statist. Sci. 23, 365--382

    2008

  29. [29]

    and Tokdar, S.T

    Martin, R. and Tokdar, S.T. (2009). Asymptotic properties of predictive recursion: robustness and rate of convergence. Electron. J. Stat. 3, 1455--1472

    2009

  30. [30]

    and Zhang, Y

    Newton, M.A., Quintana, F.A. and Zhang, Y. (1998). Nonparametric Bayes methods using predictive updating. In Practical Nonparametric and Semiparametric Bayesian Statistics, Springer

    1998

  31. [31]

    Robbins, H. (1951). Asymptotically subminimax solutions of compound decision problems. In Proceedings of the Second Berkeley Symposium 2, 131--148

    1951

  32. [32]

    Robbins, H. (1956). An empirical Bayes approach to statistics. In Proc. Third Berkeley Symp. Math. Statist. Probab. 3, 157--164

    1956

  33. [33]

    Robbins, H. (1977). Prediction and estimation for the compound Poisson distribution. Proc. Natl. Acad. Sci. U.S.A. 74 , 2670--2671

    1977

  34. [34]

    Robbins, H. (1988). The u,\,v method of estimation. In Statistical Decision Theory and Related Topics IV. Springer, New York

    1988

  35. [35]

    and Zhang, C.-H

    Robbins, H. and Zhang, C.-H. (1988). Estimating a treatment effect under biased sampling. Proc. Natl. Acad. Sci. U.S.A. 85, 3670--3672

    1988

  36. [36]

    and Zhang, C.-H

    Robbins, H. and Zhang, C.-H. (1989). Estimating the superiority of a drug to a placebo when all and only those patients at risk are treated with the drug. Proc. Natl. Acad. Sci. U.S.A. 86, 3003--3005

    1989

  37. [37]

    and Zhang, C.-H

    Robbins, H. and Zhang, C.-H. (1991). Estimating a multiplicative treatment effect under biased allocation. Biometrika 78, 349--354

    1991

  38. [38]

    and Zhang, C.-H

    Robbins, H. and Zhang, C.-H. (2000). Efficiency of the u,\,v method of estimation. Proc. Natl. Acad. Sci. U.S.A. 97, 12976--12979

    2000

  39. [39]

    M., and de Montjoye, Y

    Rocher, L., Hendrickx, J. M., and de Montjoye, Y. A. (2019). Estimating the success of re-identifications in incomplete datasets using generative models. Nat. Commun. 10, 3069

    2019

  40. [40]

    and Wu, Y

    Shen, Y. and Wu, Y. (2024). Empirical Bayes estimation: When does g -modeling beat g -modeling in theory (and in practice)? Preprint arXiv:2211.12692

    work page arXiv 2024

  41. [41]

    Skinner, and Elliot, M.J. (2002). A measure of disclosure risk for microdata. J. R. Statist. Soc. B 64, 855--867

    2002

  42. [42]

    and Makov, U.E

    Smith, A.F.M. and Makov, U.E. (1978). A quasi-Bayes sequential procedure for mixtures. J. R. Statist. Soc. B 40, 106--112

    1978

  43. [43]

    and West, M

    Tebaldi, C. and West, M. (1998). Bayesian inference on network traffic using link count data. J. Amer. Statist. Assoc. 93, 557--573

    1998

  44. [44]

    Vardi, Y. (1996). Network tomography: Estimating source-destination traffic intensities from link data. J. Amer. Statist. Assoc. 91, 365--377

    1996

  45. [45]

    Zhang, C.-H. (2005). Estimation of sums of random variables: examples and information bounds. Ann. Statist. 33, 2022--2041

    2005

  46. [46]

    Stochastic approximation and its applications

    Chen, H.F (2002). Stochastic approximation and its applications. Springer New York, NY

    2002