On adaptive posterior concentration rates

Hoffmann, Marc; Rousseau, Judith; Schmidt-Hieber, Johannes

doi:10.1214/15-aos1341

Cited by 70 publications

(115 citation statements)

References 28 publications

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“…As we have mentioned in the end of Section 5.2, the current Bayes nonparametric technique for proving posterior contraction rate only covers losses which are at the same order of Kullback-Leiber divergence. It cannot handle other non-intrinsic loss [16]. In the Bayes matrix estimation setting, whether we can show the following conclusion…”

Section: Relation To Matrix Estimation Under Non-frobenius Lossmentioning

confidence: 90%

Bernstein-von Mises theorems for functionals of the covariance matrix

Gao¹,

Zhou²

2016

Electron. J. Statist.

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We provide a general theoretical framework to derive Bernstein-von Mises theorems for matrix functionals. The conditions on functionals and priors are explicit and easy to check. Results are obtained for various functionals including entries of covariance matrix, entries of precision matrix, quadratic forms, log-determinant, eigenvalues in the Bayesian Gaussian covariance/precision matrix estimation setting, as well as for Bayesian linear and quadratic discriminant analysis. are asymptotically the same under the sampling distribution P n θ * . Note that the first one, known as the posterior, is of interest to Bayesians, and the second one is of interest to frequentists in the large sample theory. Applications of Bernstein-von Mises theorem include constructing confidence sets from Bayesian methods with frequentist coverage guarantees.Despite the success of BvM results in the classical parametric setting, little is known about the high-dimensional case, where the unknown parameter is of increasing or even infinite dimensions. The pioneering works of [11] and [13] (see also [17]) showed that generally BvM may not be true in non-classical cases. Despite the negative results, further works on some notions of nonparametric BvM provide some positive answers. See, for example, [22,8,9,24]. In this paper, we consider the question whether it is possible to have BvM results for matrix functionals, such as matrix entries and eigenvalues, when the dimension of the matrix p grows with the sample size n.This paper provides some positive answers to this question. To be specific, we consider a multivariate Gaussian likelihood and put a prior on the covariance matrix. We prove that the posterior distribution has a BvM behavior for various matrix functionals including entries of covariance matrix, entries of precision matrix, quadratic forms, log-determinant, and eigenvalues. All of these conclusions are obtained from a general theoretical framework we provide in Section 2, where we propose explicit easy-to-check conditions on both functionals and priors. We illustrate the theory by both conjugate and non-conjugate priors. A slight extension of the general framework leads to BvM results for discriminant analysis. Both linear discriminant analysis (LDA) and quadratic discriminant analysis (QDA) are considered. This work is inspired by a growing interest in studying the BvM phenomena on a lowdimensional functional of the whole parameter. That is, the asymptotic distribution of √ n(f (θ) −f )|X n , with f being a map from Θ to R d , where d does not grow with n. A special case is the semiparametric setting, where θ = (µ, η) contains both a parametric part µ and a nonparametric part η. The functional f takes the form of f (µ, η) = µ.The works in this field are pioneered by [19] in a right-censoring model and [26] for a general theory in the semiparametric setting. However, the conditions provided by [26] for BvM to hold are hard to check when specific examples are considered. To the best of our knowledge, the first general framework ...

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Section: Relation To Matrix Estimation Under Non-frobenius Lossmentioning

confidence: 90%

Bernstein-von Mises theorems for functionals of the covariance matrix

Gao¹,

Zhou²

2016

Electron. J. Statist.

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“…Under the condition of posterior concentration rate ε n (θ) at θ, E θ [r n (γ)] ε n ; see [3] for details on this result. So the only thing that remains to be verified is that lim inf…”

Section: On Polished Tail Parameters a Key Notion In This Paper Is Tmentioning

confidence: 93%

“…In [3] it is observed that when the posterior distribution has concentration rate ε n , under some loss function ℓ(·, ·), and if there exists an estimate, say,…”

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confidence: 99%

Discussion of “Frequentist coverage of adaptive nonparametric Bayesian credible sets”

Rousseau¹

2015

Ann. Statist.

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“…, X n ) iid ∼ N p (0, Σ 0 ). In the literature, the posterior is said to achieve the minimax rate if its convergence rate is the same as the frequentist minimax rate ( [36]; [21]; [29]). Since the posterior convergence rate cannot be faster than the frequentist minimax rate ( [28]), it is often called the optimal rate of posterior convergence ( [40]; [38]).…”

Section: Decision Theoretic Prior Selectionmentioning

confidence: 99%

Optimal Bayesian Minimax Rates for Unconstrained Large Covariance Matrices

Lee¹,

Lee²

2018

Bayesian Anal.

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We obtain the optimal Bayesian minimax rate for the unconstrained large covariance matrix of multivariate normal sample with mean zero, when both the sample size, n, and the dimension, p, of the covariance matrix tend to infinity. Traditionally the posterior convergence rate is used to compare the frequentist asymptotic performance of priors, but defining the optimality with it is elusive. We propose a new decision theoretic framework for prior selection and define Bayesian minimax rate. Under the proposed framework, we obtain the optimal Bayesian minimax rate for the spectral norm for all rates of p. We also considered Frobenius norm, Bregman divergence and squared log-determinant loss and obtain the optimal Bayesian minimax rate under certain rate conditions on p. A simulation study is conducted to support the theoretical results.

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On adaptive posterior concentration rates

Cited by 70 publications

References 28 publications

Bernstein-von Mises theorems for functionals of the covariance matrix

Bernstein-von Mises theorems for functionals of the covariance matrix

Discussion of “Frequentist coverage of adaptive nonparametric Bayesian credible sets”

Optimal Bayesian Minimax Rates for Unconstrained Large Covariance Matrices

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