On coverage and local radial rates of credible sets

Belitser, Eduard

doi:10.1214/16-aos1477

Cited by 36 publications

(58 citation statements)

References 26 publications

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“…Existing results cover only very special cases; see e.g. Johnstone (2010); Bontemps (2011); Panov and Spokoiny (2015); Castillo (2012); Castillo and Nickl (2013); Belitser (2017) and references therein. Most of the mentioned results are of asymptotic nature and do not quantify the accuracy of the coverage probability.…”

Section: And Hence the Vectormentioning

confidence: 99%

Large ball probabilities, Gaussian comparison and anti-concentration

Götze¹,

Naumov²,

Spokoiny³

et al. 2019

Bernoulli

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We derive tight non-asymptotic bounds for the Kolmogorov distance between the probabilities of two Gaussian elements to hit a ball in a Hilbert space. The key property of these bounds is that they are dimension-free and depend on the nuclear (Schatten-one) norm of the difference between the covariance operators of the elements and on the norm of the mean shift. The obtained bounds significantly improve the bound based on Pinsker's inequality via the Kullback-Leibler divergence. We also establish an anti-concentration bound for a squared norm of a non-centered Gaussian element in Hilbert space. The paper presents a number of examples motivating our results and applications of the obtained bounds to statistical inference and to high-dimensional CLT.

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Section: And Hence the Vectormentioning

confidence: 99%

Large ball probabilities, Gaussian comparison and anti-concentration

Götze¹,

Naumov²,

Spokoiny³

et al. 2019

Bernoulli

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“…This "deceptiveness" phenomenon is well understood for some smoothness structures (e.g., Sobolev scale), especially in global minimax settings; see [22], [9], [4] and [26]. If we now insist on the optimal size property in (2) for all Θ β , β ∈ B, the coverage relation in (2) will not hold for all Θ 0 = Θ β , but only for Θ 0 = Θ β \Θ ′ , with some set of "deceptive parameters" Θ ′ removed from Θ β .…”

Section: Introductionmentioning

confidence: 99%

“…In all the above mentioned papers global minimax radial rates (i.e., r(θ) = r(Θ β ) for all θ ∈ Θ β ) for specific smoothness structures are studied. A local approach, delivering also the adaptive minimax results for many smoothness structures simultaneously, is considered by [2] for posterior contraction rates and by [4] for constructing optimal confidence balls. In [4], yet a more general (than Θ ss and Θ pt ) set of nondeceptive parameters was introduced, Θ 0 = Θ ebr , parameters satisfying the so called excessive bias restriction (EBR).…”

Section: Introductionmentioning

confidence: 99%

Needles and straw in a haystack: Robust confidence for possibly sparse sequences

Belitser¹,

Nurushev²

2020

Bernoulli

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In the general signal+noise (allowing non-normal, non-independent observations) model we construct an empirical Bayes posterior which we then use for uncertainty quantification for the unknown, possibly sparse, signal. We introduce a novel excessive bias restriction (EBR) condition, which gives rise to a new slicing of the entire space that is suitable for uncertainty quantification. Under EBR and some mild exchangeable exponential moment condition on the noise, we establish the local (oracle) optimality of the proposed confidence ball. Without EBR, we derive the full coverage for confidence balls of at least σn 1/4 -radius, implying the local optimality only for cases when the oracle rate is at least of the order σn 1/4 . In passing, we also get the local optimal results for estimation and posterior contraction problems. Adaptive minimax results (also for the estimation and posterior contraction problems) over various sparsity classes follow from our local results. MSC2010 subject classification: primary 62G15, secondary 62C12.The best studied problem in the sparsity context is that of estimating θ in the many normal means model, a variety of estimation methods and results are available in the literature: [27]. However, even an optimal estimator does not reveal how far it is from θ. It is of importance to quantify this uncertainty, which can be seen as the problem of constructing confidence sets for θ.Bayesian approach and accompanying posterior contraction problem. Many inference methods have Bayesian connections. For example, even some seemingly non Bayesian estimators can be obtained as certain quantities (like posterior mode for penalized minimum contrast estimators) of the (empirical Bayes) posterior distributions resulting from imposing some specific priors on the parameter; cf.[17] and [1]. Although the Bayesian methodology is used or can be related to in constructing many (frequentist) inference procedures, only recently the posterior distributions themselves have been studied in the sparsity context: [13], [27], [19], [12], [7], [25], [23].In this paper, for inference on θ we use an empirical Bayes approach. Since any Bayesian approach always delivers a posterior π(ϑ|X) (in the posteriors for θ, we will use the variable ϑ to distinguish it from the "true" θ), an accompanying problem of interest is the contraction of the resulting (empirical Bayes) posterior to the "true" θ from the frequentist perspective of the "true" measure P θ , the distribution of X from (1). The quality of posterior is characterized by the posterior contraction rate. We pursue a novel local approach by allowing the posterior contraction rate to be a local quantity, i.e., depending on the true θ, whereas global minimax rates are typically studied in the literature on Bayesian nonparametrics.A common Bayesian way to model sparsity structure is by the so called two-groups priors. Such a prior puts positive mass on vectors θ with some exact zero coordinates (zero group) and the remaining coordinates (signal group) are drawn from a cho...

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“…That is, for any posterior with the optimal concentration rate, there must be true parameter values that the 100(1 − γ)% posterior credible sets cover with probability (much) less than 1 − γ. Consequently, there is interest in identifying these troublemaker parameter values. Recent efforts along these lines in the sparse normal mean model include Belitser (2017), Belitser and Nurushev (2017), van der Pas et al (2017b), Castillo and Szabó (2019), and Nurushev and Belitser (2019); for details beyond the normal mean model, see, e.g., Szabó et al (2015), Belitser and Ghosal (2019), and Martin and Tang (2019).…”

mentioning

confidence: 99%

Empirical Priors and Coverage of Posterior Credible Sets in a Sparse Normal Mean Model

Martin

Ning

2019

Sankhya A

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Bayesian methods provide a natural means for uncertainty quantification, that is, credible sets can be easily obtained from the posterior distribution. But is this uncertainty quantification valid in the sense that the posterior credible sets attain the nominal frequentist coverage probability? This paper investigates the frequentist validity of posterior uncertainty quantification based on a class of empirical priors in the sparse normal mean model. In particular, we show that our marginal posterior credible intervals achieve the nominal frequentist coverage probability under conditions slightly weaker than needed for selection consistency and a Bernstein-von Mises theorem for the full posterior, and numerical investigations suggest that our empirical Bayes method has superior frequentist coverage probability properties compared to other fully Bayes methods.

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On coverage and local radial rates of credible sets

Cited by 36 publications

References 26 publications

Large ball probabilities, Gaussian comparison and anti-concentration

Large ball probabilities, Gaussian comparison and anti-concentration

Needles and straw in a haystack: Robust confidence for possibly sparse sequences

Empirical Priors and Coverage of Posterior Credible Sets in a Sparse Normal Mean Model

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