We revisit the problem of simultaneously testing the means of @math independent normal observations under sparsity. We take a Bayesian approach to this problem by introducing a scale-mixture prior known as the normal-beta prime (NBP) prior. We first derive new concentration properties when the beta prime density is employed for a scale parameter in Bayesian hierarchical models. To detect signals in our data, we then propose a hypothesis test based on thresholding the posterior shrinkage weight under the NBP prior. Taking the loss function to be the expected number of misclassified tests, we show that our test procedure asymptotically attains the optimal Bayes risk when the signal proportion @math is known. When @math is unknown, we introduce an empirical Bayes variant of our test which also asymptotically attains the Bayes Oracle risk in the entire range of sparsity parameters @math . We further introduce restricted marginal maximum likelihood (REML) and hierarchical Bayes approaches for estimating a key hyperparameter in the NBP prior and examine multiple testing under these frameworks. Numerical experiments strongly suggest that adaptive test procedures based on REML and hierarchical Bayes considerations also have the oracle property for multiple testing. We illustrate our methodology through simulations and analysis of a prostate cancer data set.
Under certain conditions on the prior for the scale parameter @math in ), @cite_18 derived asymptotic upper bounds on Type I and Type II errors and the Bayes risk ) for both non-adaptive and data-adaptive test procedures induced by scale-mixture shrinkage priors. Under these conditions, the upper bound on the Bayes risk for scale-mixture priors is of the same order as the Bayes Oracle risk. Specifically, @cite_18 showed that the Bayes risk ) for thresholding rule ) can be bounded from above by @math for known @math and by @math for unknown @math , where @math is the constant from the fourth condition in Assumption 1 and @math are appropriate constants that depend on the prior.
@cite_18
2769906263 2094248329 1964809923 1991108537
In this paper we study the asymptotic properties of Bayesian multiple testing procedures for a large class of Gaussian scale mixture pri- ors. We study two types of multiple testing risks: a Bayesian risk proposed in (2011) where the data are assume to come from a mixture of normal, and a frequentist risk similar to the one proposed by Arias-Castro and Chen (2017). Following the work of van der (2016), we give general conditions on the prior such that both risks can be bounded. For the Bayesian risk, the bound is almost sharp. This result show that under these conditions, the considered class of continuous prior can be competitive with the usual two-group model (e.g. spike and slab priors). We also show that if the non-zeros component of the parameter are large enough, the minimax risk can be made asymptotically null. The separation rates obtained are consistent with the one that could be guessed from the existing literature (see van der , 2017b). For both problems, we then give conditions under which an adaptive version of the result can be obtained. A hierarchical Bayes Gaussian Mixture Model (HBGMM) with adaptive sparsity is proposed.We impose our hierarchical Bayes prior on the entries of precision matrices in GMM to encourage sparsity.Our adaptive sparsity can reduce the dimensionality of the parameters in GMM to be estimated.Our HBGMM method does not involve any hyperparameters to be estimated, i.e., it is hyperparameter-free. Gaussian Mixture Model (GMM) has been widely used in statistics for its great flexibility. However, parameter estimation for GMM with high dimensionality is a challenge because of the large number of parameters and the lack of observation data. In this paper, we propose an effective method named hierarchical Bayes based Adaptive Sparsity in Gaussian Mixture Model (ASGMM) to estimate the parameters in a GMM by incorporating a two-layer hierarchical Bayes based adaptive sparsity prior. The prior we impose on the precision matrices can encourage sparsity and hence reduce the dimensionality of the parameters to be estimated. In contrast to the l 1 -norm penalty or Laplace prior, our approach does not involve any hyperparameters that must be tuned, and the sparsity adapts to the observation data. The proposed method is achieved by three steps: first, we formulate an adaptive hierarchical Bayes model of the precision matrices in the GMM with a Jeffrey's noninformative hyperprior, which expresses scale-invariance and, more importantly, is hyperparameter-free and unbiased. Second, we perform a Cholesky decomposition on the precision matrices to impose the positive definite property. Finally, we exploit the expectation maximization (EM) algorithm to obtain the final estimated parameters in the GMM. Experimental results on synthetic and real-world datasets demonstrate that ASGMM cannot only adapt the sparsity of high-dimensional data with small estimated error, but also achieve better clustering performance comparing with several classical methods. This paper explores a class of empirical Bayes methods for level-dependent threshold selection in wavelet shrinkage. The prior considered for each wavelet coefficient is a mixture of an atom of probability at zero and a heavy-tailed density. The mixing weight, or sparsity parameter, for each level of the transform is chosen by marginal maximum likelihood. If estimation is carried out using the posterior median, this is a random thresholding procedure; the estimation can also be carried out using other thresholding rules with the same threshold. Details of the calculations needed for implementing the procedure are included. In practice, the estimates are quick to compute and there is software available. Simulations on the standard model functions show excellent performance, and applications to data drawn from various fields of application are used to explore the practical performance of the approach. By using a general result on the risk of the corresponding marginal maximum likelihood approach for a single sequence, overall bounds on the risk of the method are found subject to membership of the unknown function in one of a wide range of Besov classes, covering also the case of f of bounded variation. The rates obtained are optimal for any value of the parameter p in (0, ∞], simultaneously for a wide range of loss functions, each dominating the Lq norm of the σth derivative, with σ ≥ 0 and 0 < q ≤ 2. Attention is paid to the distinction between sampling the unknown function within white noise and sampling at discrete points, and between placing constraints on the function itself and on the discrete wavelet transform of its sequence of values at the observation points. Results for all relevant combinations of these scenarios are obtained. In some cases a key feature of the theory is a particular boundary-corrected wavelet basis, details of which are discussed. Overall, the approach described seems so far unique in combining the properties of fast computation, good theoretical properties and good performance in simulations and in practice. A key feature appears to be that the estimate of sparsity adapts to three different zones of estimation, first where the signal is not sparse enough for thresholding to be of benefit, second where an appropriately chosen threshold results in substantially improved estimation, and third where the signal is so sparse that the zero estimate gives the optimum accuracy rate. An empirical Bayes approach to the estimation of possibly sparse sequences observed in Gaussian white noise is set out and investigated. The prior considered is a mixture of an atom of probability at zero and a heavy-tailed density y, with the mixing weight chosen by marginal maximum likelihood, in the hope of adapting between sparse and dense sequences. If estimation is then carried out using the posterior median, this is a random thresholding procedure. Other thresholding rules employing the same threshold can also be used. Probability bounds on the threshold chosen by the marginal maximum likelihood approach lead to overall risk bounds over classes of signal sequences of length n, allowing for sparsity of various kinds and degrees. The signal classes considered are nearly black sequences where only a proportion η is allowed to be nonzero, and sequences with normalized p norm bounded by η, for η > 0 and 0 1. Simulations show excellent performance. For appropriately chosen functions y, the method is computationally tractable and software is available. The extension to a modified thresholding method relevant to the estimation of very sparse sequences is also considered.