arxiv: 2605.00455 · v1 · submitted 2026-05-01 · 📊 stat.ME · math.ST· stat.ML· stat.TH

Recognition: unknown

Concentration and Calibration in Predictive Bayesian Inference

David T. Frazier , Hui Wang

Authors on Pith no claims yet

Pith reviewed 2026-05-09 19:31 UTC · model grok-4.3

classification 📊 stat.ME math.STstat.MLstat.TH

keywords predictive Bayesian inferencecredible setscoveragecalibrationposterior concentrationforward predictive modelmodel misspecificationdata-generating process

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The pith

If the forward predictive model misses key data features, predictive Bayesian credible sets can have coverage arbitrarily close to zero.

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

Predictive Bayesian inference obtains uncertainty statements for a functional of interest solely by specifying a forward model for future data, without requiring a full likelihood or prior. The paper establishes that the resulting posterior concentrates on a limit point fully determined by this predictive model and that the credible sets inherit their coverage properties from the same source. When the predictive model fails to capture relevant aspects of the data, even elementary examples produce credible sets whose actual frequentist coverage of the true functional value can approach zero. The finding matters because it shows that the apparent flexibility of PBI comes at the cost of calibration unless the predictive engine matches the true data-generating process in a precise sense.

Core claim

When predictive Bayesian inference is implemented through a forward predictive model, the posterior for a population functional concentrates onto a well-defined quantity that depends explicitly on the predictive model. The uncertainty quantification produced by the method is likewise entirely governed by this model. Consequently, if the predictive model does not contain the true data-generating process, the frequentist coverage of the resulting credible sets for the population value of the functional can be made arbitrarily close to zero, even in simple settings. Calibration of the inferences therefore requires that the predictive engine contain the true DGP in a well-defined sense.

What carries the argument

The forward predictive model used to implement the predictive recursion, which generates the future observations that determine both the concentration point of the posterior and the coverage behavior of the credible sets.

If this is right

The posterior concentrates onto a quantity that depends explicitly on the chosen forward predictive model.
Uncertainty quantification in PBI is completely determined by the predictive model.
Coverage of credible sets for the population functional can approach zero whenever the predictive model misses relevant data features.
Calibrated posterior inferences require the predictive engine to contain the true data-generating process.

Where Pith is reading between the lines

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

Users of PBI should first check whether their forward predictive model reproduces key statistical features of the observed data before relying on the credible sets.
The result suggests comparing PBI outputs against standard Bayesian or frequentist procedures in settings where model misspecification is suspected.
Diagnostic tools that quantify how well the predictive model captures the data-generating process could be developed to flag when coverage is likely to be poor.
In complex or high-dimensional applications, ensuring the predictive model contains the true DGP may require hybrid modeling strategies that blend predictive recursion with partial likelihood information.

Load-bearing premise

The forward predictive model accurately reflects the true data-generating process for the functional of interest.

What would settle it

In one of the paper's simple examples, compute the actual frequentist coverage of the predictive Bayes credible sets under an intentionally misspecified predictive model and find that it remains close to the nominal level rather than approaching zero.

Figures

Figures reproduced from arXiv: 2605.00455 by David T. Frazier, Hui Wang.

**Figure 1.** Figure 1: Simulation paths with mean and variance biases in the predictive engine. The dotted line view at source ↗

**Figure 2.** Figure 2: Posterior contours for mean and variance functionals under different variance-bias choices view at source ↗

**Figure 3.** Figure 3: PPC-PE results by model and discrepancy, view at source ↗

**Figure 4.** Figure 4: This figure gives the same information as Figure view at source ↗

**Figure 5.** Figure 5: Simulation paths of the Gaussian predictive engine for various levels of variance bias. view at source ↗

**Figure 6.** Figure 6: Representative jellyfish of the QMP at the view at source ↗

**Figure 7.** Figure 7: The contour of estimated parameters in treatment 2 and treatment 3. view at source ↗

read the original abstract

Predictive Bayesian inference (PBI) represents a model-and prior-agnostic approach to standard Bayesian inference which allows users to quantify uncertainty for a functional of interest only by specifying a forward predictive model for future unobserved data. The flexibility and generality of this framework have led to a host of novel algorithms for implementing this approach, and many empirical applications, yet the reliability of the resulting inferences for the underlying statistical functional of interest remains unclear. Herein, we demonstrate that when using PBI for a population functional of interest, the resulting posterior concentrates onto a well-defined quantity that explicitly depends on the forward predictive model used to implement the predictive recursion underlying the method. Furthermore, the forward predictive model entirely determines the uncertainty quantification produced in PBI. Consequently, our results show that if the predictive model does not capture all relevant features of the data, and, even in very simple examples, the coverage of predictive Bayes credible sets for the population value of the functional of interest can be arbitrarily close to zero. We carefully explain why this occurs, and show that this behavior is directly tied to the inaccuracy of the forward predictive model used to produce future observations within the PBI framework. As a consequence, our results imply that in order for PBI to deliver calibrated posterior inferences, the resulting predictive engine used to generate posterior samples must contain, in a well-defined sense, the true DGP, else inferences generated under this framework will not be calibrated.

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.

Desk Editor's Note private letter to a colleague

PBI posteriors concentrate on a forward-model quantity, so credible-set coverage for population functionals can drop near zero under misspecification.

read the letter

The main thing to know is that this paper shows predictive Bayesian inference can produce credible sets whose coverage for the true population functional is arbitrarily bad when the forward predictive model is misspecified. The posterior concentrates on a quantity fixed entirely by the predictive recursion, and the uncertainty quantification inherits that dependence, which the authors link directly to the model's failure to capture relevant data features. Even simple examples illustrate the coverage collapse. This follows from the method's construction rather than from any hidden fitting step, so the argument is internally consistent once the recursion is accepted. They explain the mechanism plainly and avoid overclaiming generality beyond the stated conditions. That is useful because PBI is promoted for its model- and prior-agnostic flexibility, yet the calibration requirement turns out to be stricter than many users might assume. The paper earns credit for making the concentration explicit and for tying the coverage failure to the predictive engine's inaccuracy without circularity. Soft spots are mostly about scope and practicality. The results rest on the forward model not containing the true DGP in the required sense, but the paper does not quantify how often this occurs in typical applications or offer diagnostics that would flag the problem before inference. The technical conditions for the concentration result are stated but would need checking for edge cases or relaxations. The claim that the predictive model must contain the true DGP is strong; whether weaker forms of containment or post-hoc adjustments could restore calibration is left open. This work is for people already using or extending predictive recursion methods in statistics and machine learning, especially those concerned with uncertainty quantification for functionals. A reader who wants to understand the limits of PBI will get a clear cautionary result. It deserves a serious referee because the core derivation is grounded in the method's own structure and the limitation is practically relevant, even if the paper stops short of fixes or extensive simulations.

Referee Report

1 major / 1 minor

Summary. The paper claims that in Predictive Bayesian Inference (PBI), which relies on a forward predictive model for future data to quantify uncertainty about a functional of interest, the resulting posterior concentrates on a well-defined quantity fully determined by that predictive model. It shows that under misspecification of the forward model, the coverage of PBI credible sets for the true population value of the functional can be arbitrarily close to zero, even in simple examples, and that this behavior is tied directly to the inaccuracy of the predictive engine. The authors conclude that calibrated inferences require the predictive model to contain the true data-generating process in a well-defined sense.

Significance. If the derivations hold, the result is significant for highlighting a key limitation of PBI under the misspecification that is routine in applications. The explicit identification of the concentration target and the direct link to predictive-model inaccuracy provide a clear theoretical warning that could inform both methodological refinements and practical use of PBI algorithms. The emphasis on rigorous results and the tie to calibration requirements adds value to the literature on predictive and Bayesian methods.

major comments (1)

The central concentration and coverage claims rest on properties of the predictive recursion; a precise statement of the main theorem (including all regularity conditions on the forward model and the functional) is needed to evaluate the scope and to confirm that the arbitrarily poor coverage result is not an artifact of a narrow class of misspecifications.

minor comments (1)

The manuscript would benefit from an explicit numerical illustration (even a simple one) showing coverage approaching zero, to complement the theoretical argument and make the practical implication more concrete.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of the paper's significance and for the constructive suggestion to strengthen the presentation of our main result. We address the comment below and will incorporate the requested changes in the revised manuscript.

read point-by-point responses

Referee: The central concentration and coverage claims rest on properties of the predictive recursion; a precise statement of the main theorem (including all regularity conditions on the forward model and the functional) is needed to evaluate the scope and to confirm that the arbitrarily poor coverage result is not an artifact of a narrow class of misspecifications.

Authors: We agree that a fully precise statement of the main theorem, with explicit regularity conditions, is necessary to delineate the result's scope. In the revision we will add a formal theorem statement that lists all conditions on the forward predictive model (e.g., continuity of the predictive density in total variation, uniform integrability of the relevant moments) and on the functional of interest (e.g., continuity and boundedness with respect to the weak topology). The theorem will be stated for a general class of misspecifications rather than isolated examples, and a short remark will explain why the concentration target and the resulting coverage failure follow directly from the predictive recursion under these conditions, thereby showing that the poor coverage is not an artifact of narrow cases. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper derives its central concentration result—that the PBI posterior concentrates onto a quantity explicitly determined by the forward predictive model—from the mathematical properties of the predictive recursion construction itself. This is a direct consequence of the framework's definition and does not reduce to any fitted parameters from the evaluation data, self-definitional loops, or load-bearing self-citations. The demonstration of potential zero coverage under misspecification follows logically from analyzing the model's inaccuracy without circular reduction, and the calibration requirement is stated as an external condition on the predictive engine. The derivation remains self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; full technical assumptions on the predictive recursion and data-generating process are not available for detailed audit.

axioms (1)

standard math Standard regularity conditions on the predictive recursion and posterior concentration in Bayesian inference
Invoked implicitly to establish concentration onto a well-defined quantity.

pith-pipeline@v0.9.0 · 5553 in / 1147 out tokens · 32877 ms · 2026-05-09T19:31:05.288577+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

223 extracted references · 50 canonical work pages · 1 internal anchor

[1]

arXiv preprint arXiv:1603.02485 , year=

The block pseudo-marginal sampler , author=. arXiv preprint arXiv:1603.02485 , year=

work page arXiv
[2]

Annals of the Institute of Statistical Mathematics , volume=

The geometry of proper scoring rules , author=. Annals of the Institute of Statistical Mathematics , volume=. 2007 , publisher=

2007
[3]

Frazier, Jeremias Knoblauch, and Susan Wei

TabMGP: Martingale Posterior with TabPFN , author=. arXiv preprint arXiv:2510.25154 , year=

work page arXiv
[4]

Martingale posterior neural processes

Martingale posterior neural processes , author=. arXiv preprint arXiv:2304.09431 , year=

work page arXiv
[5]

Biometrika , pages=

Asymptotics for a class of parametric martingale posteriors , author=. Biometrika , pages=. 2026 , publisher=

2026
[6]

Annals of Statistics , pages=

Entropies and rates of convergence for maximum likelihood and Bayes estimation for mixtures of normal densities , author=. Annals of Statistics , pages=. 2001 , publisher=

2001
[7]

arXiv preprint arXiv:2406.05213 , year=

On subjective uncertainty quantification and calibration in natural language generation , author=. arXiv preprint arXiv:2406.05213 , year=

work page arXiv
[8]

Electronic journal of statistics , volume=

Reproducible parameter inference using bagged posteriors , author=. Electronic journal of statistics , volume=
[9]

Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

A general framework for updating belief distributions , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 2016 , publisher=

2016
[10]

Nature , volume=

Accurate predictions on small data with a tabular foundation model , author=. Nature , volume=. 2025 , publisher=

2025
[11]

arXiv preprint arXiv:2403.16828 , year=

Asymptotics of predictive distributions driven by sample means and variances , author=. arXiv preprint arXiv:2403.16828 , year=

work page arXiv
[12]

International conference on artificial intelligence and statistics , pages=

Posterior uncertainty quantification in neural networks using data augmentation , author=. International conference on artificial intelligence and statistics , pages=. 2024 , organization=

2024
[13]

arXiv preprint arXiv:2602.04596 , year=

A principled framework for uncertainty decomposition in TabPFN , author=. arXiv preprint arXiv:2602.04596 , year=

work page arXiv
[14]

, month = mar, year =

Yang, Yun and Dunson, David B. , month = mar, year =. Minimax. doi:10.48550/arXiv.1403.1345 , abstract =

work page doi:10.48550/arxiv.1403.1345
[15]

TEST , author =

Objective. TEST , author =. 2019 , pages =. doi:10.1007/s11749-018-0597-z , language =

work page doi:10.1007/s11749-018-0597-z 2019
[16]

Detecting Model Misspecification in Bayesian Inverse Problems via Variational Gradient Descent

Liu, Qingyang and Fisher, Matthew A. and Shen, Zheyang and Tant, Katy and Zhao, Xuebin and Curtis, Andrew and Oates, Chris J. , month = dec, year =. Detecting. doi:10.48550/arXiv.2512.01667 , abstract =

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2512.01667
[17]

Journal of Environmental Economics and Management , author =

On the. Journal of Environmental Economics and Management , author =. 1996 , pages =. doi:10.1006/jeem.1996.0052 , language =

work page doi:10.1006/jeem.1996.0052 1996
[18]

Journal of Environmental Economics and Management , author =

Hedonic housing prices and the demand for clean air , volume =. Journal of Environmental Economics and Management , author =. 1978 , pages =. doi:10.1016/0095-0696(78)90006-2 , language =

work page doi:10.1016/0095-0696(78)90006-2 1978
[19]

Sharrock, Louis and Nemeth, Christopher , month = oct, year =. Tuning-. doi:10.48550/arXiv.2510.25315 , abstract =

work page doi:10.48550/arxiv.2510.25315
[20]

Uncertainty in

Uncertainty in. Bayesian Analysis , author =. doi:10.1214/25-BA1569 , number =

work page doi:10.1214/25-ba1569
[21]

The Annals of Statistics , pages=

Proper local scoring rules , author=. The Annals of Statistics , pages=. 2012 , publisher=

2012
[22]

Hu, Kaitong and Ren, Zhenjie and Siska, David and Szpruch, Lukasz , month = dec, year =. Mean-. doi:10.48550/arXiv.1905.07769 , abstract =

work page doi:10.48550/arxiv.1905.07769 1905
[23]

Advances in Neural Information Processing Systems , volume=

Learning under model misspecification: Applications to variational and ensemble methods , author=. Advances in Neural Information Processing Systems , volume=
[24]

The Annals of Statistics , author =

Gibbs posterior for variable selection in high-dimensional classification and data mining , volume =. The Annals of Statistics , author =. 2008 , file =. doi:10.1214/07-AOS547 , number =

work page doi:10.1214/07-aos547 2008
[25]

The Annals of Statistics , author =

From -entropy to. The Annals of Statistics , author =. doi:10.1214/009053606000000704 , number =

work page doi:10.1214/009053606000000704
[26]

IEEE Transactions on Information Theory , author =

Information-theoretic upper and lower bounds for statistical estimation , volume =. IEEE Transactions on Information Theory , author =. 2006 , pages =. doi:10.1109/TIT.2005.864439 , number =

work page doi:10.1109/tit.2005.864439 2006
[27]

Electronic Journal of Statistics , author =

The. Electronic Journal of Statistics , author =. 2012 , file =. doi:10.1214/12-EJS675 , number =

work page doi:10.1214/12-ejs675 2012
[28]

Martin, Ryan and Syring, Nicholas , year =. Direct. Handbook of. doi:10.1016/bs.host.2022.06.004 , pages =

work page doi:10.1016/bs.host.2022.06.004 2022
[29]

The American Statistician , author =

Optimal Information Processing and. The American Statistician , author =. 1988 , pages =. doi:10.2307/2685143 , number =

work page doi:10.2307/2685143 1988
[30]

M., Hill A

palmerpenguins: Palmer Archipelago (Antarctica) penguin data , author =. 2020 , note =. doi:10.5281/zenodo.3960218 , url =

work page doi:10.5281/zenodo.3960218 2020
[31]

2020 , eprint=

The Case for Bayesian Deep Learning , author=. 2020 , eprint=

2020
[32]

2025 , eprint=

Improved Finite-Particle Convergence Rates for Stein Variational Gradient Descent , author=. 2025 , eprint=

2025
[33]

Proceedings of The 25th International Conference on Artificial Intelligence and Statistics , pages =

Convex Analysis of the Mean Field Langevin Dynamics , author =. Proceedings of The 25th International Conference on Artificial Intelligence and Statistics , pages =. 2022 , editor =

2022
[34]

Mixture Models: Theory, Geometry, and Applications , author =
[35]

Journal of the American Statistical Association , publisher =

Nonparametric maximum likelihood estimation of a mixing distribution , author =. Journal of the American Statistical Association , publisher =
[36]

Ecole d’Et

Topics in non-parametric statistics , author=. Ecole d’Et
[37]

The Annals of Statistics , pages=

Aggregation for Gaussian Regression , author=. The Annals of Statistics , pages=. 2007 , publisher=

2007
[38]

Advances in Neural Information Processing Systems , volume=

Convergence of mean-field langevin dynamics: time-space discretization, stochastic gradient, and variance reduction , author=. Advances in Neural Information Processing Systems , volume=
[39]

2023 , booktitle=

Mean-field Underdamped Langevin Dynamics and its Spacetime Discretization , author=. 2023 , booktitle=

2023
[40]

Polyak--

Liu, Linshan and Majka, Mateusz B and Szpruch,. Polyak--. Applied Mathematics & Optimization , volume=. 2023 , publisher=

2023
[41]

arXiv preprint arXiv:2510.01358 , year=

A theoretical framework for M-posteriors: frequentist guarantees and robustness properties , author=. arXiv preprint arXiv:2510.01358 , year=

work page arXiv
[42]

Convex Optimization over Probability Measures , author =
[43]

Journal of Machine Learning Research , author =

Model. Journal of Machine Learning Research , author =. 2022 , pages =

2022
[44]

Bayesian

Wilson, Andrew G and Izmailov, Pavel , editor =. Bayesian. Advances in. 2020 , pages =

2020
[45]

Bayesian Analysis , author =

Bayesian. Bayesian Analysis , author =. 2023 , file =. doi:10.1214/22-BA1311 , number =

work page doi:10.1214/22-ba1311 2023
[46]

Bayesian Analysis , author =

Cross-. Bayesian Analysis , author =. doi:10.1214/23-BA1409 , number =

work page doi:10.1214/23-ba1409
[47]

Journal of Statistical Computation and Simulation , author =

Approximate leave-future-out cross-validation for. Journal of Statistical Computation and Simulation , author =. 2020 , pages =. doi:10.1080/00949655.2020.1783262 , language =

work page doi:10.1080/00949655.2020.1783262 2020
[48]

Bayesian Analysis , volume=

Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It , author=. Bayesian Analysis , volume=
[49]

Biometrika , author =

Valid sequential inference on probability forecast performance , volume =. Biometrika , author =. 2022 , note =. doi:10.1093/biomet/asab047 , abstract =

work page doi:10.1093/biomet/asab047 2022
[50]

Journal of the Royal Statistical Society Series B: Statistical Methodology , author =

Isotonic. Journal of the Royal Statistical Society Series B: Statistical Methodology , author =. 2021 , note =. doi:10.1111/rssb.12450 , abstract =

work page doi:10.1111/rssb.12450 2021
[51]

Teaching Statistics , author =

A. Teaching Statistics , author =. 2002 , note =. doi:10.1111/1467-9639.00097 , abstract =

work page doi:10.1111/1467-9639.00097 2002
[52]

1995 , organization=

Mixture models: theory, geometry, and applications , author=. 1995 , organization=

1995
[53]

Journal of the American Statistical Association , volume=

Nonparametric maximum likelihood estimation of a mixing distribution , author=. Journal of the American Statistical Association , volume=. 1978 , publisher=

1978
[54]

Biometrika , author =

Universal robust regression via maximum mean discrepancy , volume =. Biometrika , author =. 2024 , note =. doi:10.1093/biomet/asad031 , abstract =

work page doi:10.1093/biomet/asad031 2024
[55]

Steven L

Statistical. Statistical Science , author =. 2001 , file =. doi:10.1214/ss/1009213726 , number =

work page doi:10.1214/ss/1009213726 2001
[56]

Journal of the American Statistical Association , year=

Robust Bayesian inference via coarsening , author=. Journal of the American Statistical Association , year=
[57]

Biometrika , volume=

Goodness of prediction fit , author=. Biometrika , volume=. 1975 , publisher=

1975
[58]

, author=

Universality, Characteristic Kernels and RKHS Embedding of Measures. , author=. Journal of Machine Learning Research , volume=
[59]

, author=

Asymptotic Normality, Concentration, and Coverage of Generalized Posteriors. , author=. J. Mach. Learn. Res. , volume=
[60]

Advances in neural information processing systems , volume=

Generative adversarial nets , author=. Advances in neural information processing systems , volume=
[61]

arXiv preprint arXiv:2012.04505 , year=

Gibbs posterior concentration rates under sub-exponential type losses , author=. arXiv preprint arXiv:2012.04505 , year=

work page arXiv 2012
[62]

Journal of the American Statistical Association , volume=

Metropolis--Hastings via classification , author=. Journal of the American Statistical Association , volume=. 2023 , publisher=

2023
[63]

Statistics and Computing , volume=

Noisy Monte Carlo: Convergence of Markov chains with approximate transition kernels , author=. Statistics and Computing , volume=. 2016 , publisher=

2016
[64]

Robust and Scalable

Altamirano, Matias and Briol, Francois-Xavier and Knoblauch, Jeremias , booktitle=. Robust and Scalable
[65]

International Conference on Machine Learning , pages=

Automatic posterior transformation for likelihood-free inference , author=. International Conference on Machine Learning , pages=. 2019 , organization=

2019
[66]

Advances in neural information processing systems , volume=

Flexible statistical inference for mechanistic models of neural dynamics , author=. Advances in neural information processing systems , volume=
[67]

Bernoulli , volume=

Gibbs posterior concentration rates under sub-exponential type losses , author=. Bernoulli , volume=. 2023 , publisher=

2023
[68]

The 22nd International Conference on Artificial Intelligence and Statistics , pages=

Sequential neural likelihood: Fast likelihood-free inference with autoregressive flows , author=. The 22nd International Conference on Artificial Intelligence and Statistics , pages=. 2019 , organization=

2019
[69]

Robust and Conjugate

Altamirano, Matias and Briol, Fran. Robust and Conjugate. arXiv preprint arXiv:2311.00463 , year=

work page arXiv
[70]

Biometrika , volume=

A semiparametric estimation procedure of dependence parameters in multivariate families of distributions , author=. Biometrika , volume=. 1995 , publisher=

1995
[71]

Dellaporta, Charita and Knoblauch, Jeremias and Damoulas, Theodoros and Briol, Fran. Robust. International Conference on Artificial Intelligence and Statistics , pages=. 2022 , organization=

2022
[72]

Statistics and Computing , volume=

Numerical maximum likelihood estimation for the g-and-k and generalized g-and-h distributions , author=. Statistics and Computing , volume=. 2002 , publisher=

2002
[73]

Journal of the American Statistical Association , volume=

Estimation of copulas via maximum mean discrepancy , author=. Journal of the American Statistical Association , volume=. 2023 , publisher=

2023
[74]

Likelihood-free

Drovandi, Christopher C and Pettitt, Anthony N , journal=. Likelihood-free. 2011 , publisher=

2011
[75]

2009 , publisher=

Allingham, David and King, Robert AR and Mengersen, Kerrie L , journal=. 2009 , publisher=

2009
[76]

Biometrika , volume=

Robust and efficient estimation by minimising a density power divergence , author=. Biometrika , volume=. 1998 , publisher=

1998
[77]

Ghosh, Abhik and Basu, Ayanendranath , journal=. Robust. 2016 , publisher=

2016
[78]

Advances in applied probability , volume=

Integral probability metrics and their generating classes of functions , author=. Advances in applied probability , volume=. 1997 , publisher=

1997
[79]

On the empirical estimation of integral probability metrics , author=
[80]

The Journal of Machine Learning Research , volume=

A kernel two-sample test , author=. The Journal of Machine Learning Research , volume=. 2012 , publisher=

2012

Showing first 80 references.