Optimal Bayesian Minimax Rates for Unconstrained Large Covariance Matrices

Lee, Kyoungjae; Lee, Jae Yong

doi:10.1214/18-ba1094

Cited by 20 publications

(31 citation statements)

References 30 publications

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“…for all sufficiently large n and some constant C λ > 0, where the last inequality follows from Lemma B.7 in Lee and Lee (2018). Also note that (n −1 X T n X n ) −1 P (j) 0,l − (Σ 0,P (j) 0,l…”

Section: Appendix A: Proofs Of Main Theoremsmentioning

confidence: 91%

“…then E π ((Σ P (j) 0,l on the eventÑ nj (C λ ) c , for some positive constants c 1 and c 2 depending only on C λ . We note here that we are using different parametrization for Wishart and inverse Wishart distributions compared to Lee and Lee (2018). Moreover, by Lemma B.7 in Lee and Lee (2018) and Condition (B4),…”

Section: Appendix A: Proofs Of Main Theoremsmentioning

confidence: 99%

“…Especially, a precision matrix reveals the conditional dependences between the variables. However, especially when the number of variables p can be much larger than the sample size n, it is a challenging task because a consistent estimation is impossible without further assumptions (Lee and Lee, 2018).…”

Section: Introductionmentioning

confidence: 99%

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Bayesian inference for high-dimensional decomposable graphs

Lee

Cao

2021

Electron. J. Statist.

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In this paper, we consider high-dimensional Gaussian graphical models where the true underlying graph is decomposable. A hierarchical G-Wishart prior is proposed to conduct a Bayesian inference for the precision matrix and its graph structure. Although the posterior asymptotics using the G-Wishart prior has received increasing attention in recent years, most of the results assume moderate high-dimensional settings, where the number of variables p is smaller than the sample size n. However, this assumption might not hold in many real applications such as genomics, speech recognition and climatology. Motivated by this gap, we investigate asymptotic properties of posteriors under the high-dimensional setting where p can be much larger than n. The pairwise Bayes factor consistency, posterior ratio consistency and graph selection consistency are obtained in this high-dimensional setting. Furthermore, the posterior convergence rate for precision matrices under the matrix 1 -norm is derived, which is faster than posterior convergence rates obtained in existing literature. A simulation study confirms that the proposed Bayesian procedure outperforms competitors.

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Section: Appendix A: Proofs Of Main Theoremsmentioning

confidence: 91%

Section: Appendix A: Proofs Of Main Theoremsmentioning

confidence: 99%

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Bayesian inference for high-dimensional decomposable graphs

Lee

Cao

2021

Electron. J. Statist.

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“…(iii) If γ(k) = Ck −α for some constants α > 0 and C > 0, then we have inf Ωn sup Ω 0,n ∈U ( 0 ,γ)E 0n Ω n − Ω 0Remark Since a frequentist minimax lower bound is also a P-loss minimax lower bound, Theorem 3.1 implies a P-loss minimax lower bound. For the proof of this argument, seeLee and Lee (2017).To the best of our knowledge, there is no frequentist minimax lower bound result on this setting. The estimation of precision matrix with polynomially banded Cholesky factor under the spectral norm was studied by Bickel and Levina (2008b), but they did not consider the minimax lower bound.…”

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confidence: 94%

Estimating large precision matrices via modified Cholesky decomposition

Lee¹,

Lee²

2020

STAT SINICA

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We introduce the k-banded Cholesky prior for estimating a high-dimensional bandable precision matrix via the modified Cholesky decomposition. The bandable assumption is imposed on the Cholesky factor of the decomposition. We obtained the P-loss convergence rate under the spectral norm and the matrix ∞ norm and the minimax lower bounds. Since the P-loss convergence rate (Lee and Lee (2017)) is stronger than the posterior convergence rate, the rates obtained are also posterior convergence rates. Furthermore, when the true precision matrix is a k 0 -banded matrix with some finite k 0 , the obtained P-loss convergence rates coincide with the minimax rates. The established convergence rates are slightly slower than the minimax lower bounds, but these are the fastest rates for bandable precision matrices among the existing Bayesian approaches. A simulation study is conducted to compare the performance to the other competitive estimators in various scenarios.

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“…Consider an empirical Bayes approach by considering priors (4) and (5) with k instead of imposing a prior on k. This empirical Bayes method faciliates easy implementations when the estimation of Cholesky factor or precision matrix is of interest. To assess the performance, we adopt the P-loss convergence rate used by Castillo (2014) and Lee and Lee (2018). Corollary .1 presents the P-loss convergence rate of the empirical Bayes approach with respect to the Cholesky factor under the matrix ∞ -norm.…”

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confidence: 99%

Bayesian Bandwidth Test and Selection for High-dimensional Banded Precision Matrices

Lee¹,

Lin²

2020

Bayesian Anal.

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Assuming a banded structure is one of the common practice in the estimation of highdimensional precision matrix. In this case, estimating the bandwidth of the precision matrix is a crucial initial step for subsequent analysis. Although there exist some consistent frequentist tests for the bandwidth parameter, bandwidth selection consistency for precision matrices has not been established in a Bayesian framework. In this paper, we propose a prior distribution tailored to the bandwidth estimation of high-dimensional precision matrices. The banded structure is imposed via the Cholesky factor from the modified Cholesky decomposition. We establish the strong model selection consistency for the bandwidth as well as the consistency of the Bayes factor. The convergence rates for Bayes factors under both the null and alternative hypotheses are derived which yield similar order of rates. As a by-product, we also proposed an estimation procedure for the Cholesky factors yielding an almost optimal order of convergence rates. Two-sample bandwidth test is also considered, and it turns out that our method is able to consistently detect the equality of bandwidths between two precision matrices. The simulation study confirms that our method in general outperforms the existing frequentist and Bayesian methods.

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Optimal Bayesian Minimax Rates for Unconstrained Large Covariance Matrices

Cited by 20 publications

References 30 publications

Bayesian inference for high-dimensional decomposable graphs

Bayesian inference for high-dimensional decomposable graphs

Estimating large precision matrices via modified Cholesky decomposition

Bayesian Bandwidth Test and Selection for High-dimensional Banded Precision Matrices

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