arXiv subjects

Ray Bai

Publications and source records attributed to Ray Bai.

22 records · Page 2Linked to original sources

arXiv2019

Large-Scale Multiple Hypothesis Testing with the Normal-Beta Prime Prior

We revisit the problem of simultaneously testing the means of $n$ 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 $p$ is known. When $p$ 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 $p \propto n^{-ε}, ε\in (0, 1)$. Finally, we also consider 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.

stat.ME↗

arXiv2019

On the Beta Prime Prior for Scale Parameters in High-Dimensional Bayesian Regression Models

We study high-dimensional Bayesian linear regression with a general beta prime distribution for the scale parameter. Under the assumption of sparsity, we show that appropriate selection of the hyperparameters in the beta prime prior leads to the (near) minimax posterior contraction rate when $p \gg n$. For finite samples, we propose a data-adaptive method for estimating the hyperparameters based on marginal maximum likelihood (MML). This enables our prior to adapt to both sparse and dense settings, and under our proposed empirical Bayes procedure, the MML estimates are never at risk of collapsing to zero. We derive efficient Monte Carlo EM and variational EM algorithms for implementing our model, which are available in the R package NormalBetaPrime. Simulations and analysis of a gene expression data set illustrate our model's self-adaptivity to varying levels of sparsity and signal strengths.

stat.ME↗

arXiv2018

The Inverse Gamma-Gamma Prior for Optimal Posterior Contraction and Multiple Hypothesis Testing

We study the well-known problem of estimating a sparse $n$-dimensional unknown mean vector $θ= (θ_1, ..., θ_n)$ with entries corrupted by Gaussian white noise. In the Bayesian framework, continuous shrinkage priors which can be expressed as scale-mixture normal densities are popular for obtaining sparse estimates of $θ$. In this article, we introduce a new fully Bayesian scale-mixture prior known as the inverse gamma-gamma (IGG) prior. We prove that the posterior distribution contracts around the true $θ$ at (near) minimax rate under very mild conditions. In the process, we prove that the sufficient conditions for minimax posterior contraction given by Van der Pas et al. (2016) are not necessary for optimal posterior contraction. We further show that the IGG posterior density concentrates at a rate faster than those of the horseshoe or the horseshoe+ in the Kullback-Leibler (K-L) sense. To classify true signals ($θ_i \neq 0$), we also propose a hypothesis test based on thresholding the posterior mean. Taking the loss function to be the expected number of misclassified tests, we show that our test procedure asymptotically attains the optimal Bayes risk exactly. We illustrate through simulations and data analysis that the IGG has excellent finite sample performance for both estimation and classification.

math.ST↗

arXiv2018

High-Dimensional Multivariate Posterior Consistency Under Global-Local Shrinkage Priors

We consider sparse Bayesian estimation in the classical multivariate linear regression model with $p$ regressors and $q$ response variables. In univariate Bayesian linear regression with a single response $y$, shrinkage priors which can be expressed as scale mixtures of normal densities are popular for obtaining sparse estimates of the coefficients. In this paper, we extend the use of these priors to the multivariate case to estimate a $p \times q$ coefficients matrix $\mathbf{B}$. We derive sufficient conditions for posterior consistency under the Bayesian multivariate linear regression framework and prove that our method achieves posterior consistency even when $p>n$ and even when $p$ grows at nearly exponential rate with the sample size. We derive an efficient Gibbs sampling algorithm and provide the implementation in a comprehensive R package called MBSP. Finally, we demonstrate through simulations and data analysis that our model has excellent finite sample performance.

stat.ME↗