Large-scale multiple hypothesis testing with the normal-beta prime prior

Abstract

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 studying a scale-mixture prior known as the normal-beta prime (NBP) prior. To detect signals, we 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 ∝ n − , ∈ (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.

Horseshoe Regularisation for Machine Learning in Complex and Deep Models1

“…Datta and Ghosh (2013) proved that the decision rule induced by the horseshoe estimator is asymptotically Bayes optimal for multiple testing under 0-1 loss up to a multiplicative constant. This result was generalized to include other global-local priors by Ghosh et al (2016) and Bai and Ghosh (2018), among others. Pas et al (2014) showed the horseshoe estimator is minimax in 2 in a nearly-black case up to a constant.…”

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

“…p, is unbounded with a singularity at zero for any 0 < a ≤ 1/2. [3]. Proposition 2.1 implies that in order to facilitate sparse recovery of β, we should set the hyperparameter a to be a small value.…”

“…We call our model the normal-beta prime (NBP) prior. Bai and Ghosh [3] previously studied the NBP model in the context of multiple hypothesis testing of normal means. Here, we extend the NBP prior to high-dimensional linear regression (1.1).…”

Bayesian spatial and spatiotemporal models based on multiscale factorizations

“…The decision to threshold wavelet coefficients to zero may be seen as a form of variable selection with respect to the wavelet basis representation (e.g., see Stingo, Vannucci, & Downey, 2012). Therefore, other recent priors for Bayesian variable selection such as the horseshoe prior (Carvalho, Polson, & Scott, 2010), the Dirichlet–Laplace prior (Bhattacharya, Pati, Pillai, & Dunson, 2015), and the normal‐beta prime prior (Bai & Ghosh, 2019) may be considered as priors for wavelet coefficients. We note that wavelet representations of signals are usually sparse, that is, few wavelet coefficients contain most of the information and thus many wavelet coefficients may be set to zero without much loss of information.…”