We propose a new prior for ultra-sparse signal detection that we term the “horseshoe+ prior.” The horseshoe+ prior is a natural extension of the horseshoe prior that has achieved success in the estimation and detection of sparse signals and has been shown to possess a number of desirable theoretical properties while enjoying computational feasibility in high dimensions. The horseshoe+ prior builds upon these advantages. Our work proves that the horseshoe+ posterior concentrates at a rate faster than that of the horseshoe in the Kullback–Leibler (K-L) sense. We also establish theoretically that the proposed estimator has lower posterior mean squared error in estimating signals compared to the horseshoe and achieves the optimal Bayes risk in testing up to a constant. For one-group global–local scale mixture priors, we develop a new technique for analyzing the marginal sparse prior densities using the class of Meijer-G functions. In simulations, the horseshoe+ estimator demonstrates superior performance in a standard design setting against competing methods, including the horseshoe and Dirichlet–Laplace estimators. We conclude with an illustration on a prostate cancer data set and by pointing out some directions for future research.

Enteropathy-associated T cell lymphoma (EATL) is a lethal, and the most common, neoplastic complication of celiac disease. Here, we defined the genetic landscape of EATL through whole-exome sequencing of 69 EATL tumors. SETD2 was the most frequently silenced gene in EATL (32% of cases). The JAK-STAT pathway was the most frequently mutated pathway, with frequent mutations in STAT5B as well as JAK1, JAK3, STAT3, and SOCS1. We also identified mutations in KRAS, TP53, and TERT. Type I EATL and type II EATL (monomorphic epitheliotropic intestinal T cell lymphoma) had highly overlapping genetic alterations indicating shared mechanisms underlying their pathogenesis. We modeled the effects of SETD2 loss in vivo by developing a T cell–specific knockout mouse. These mice manifested an expansion of γδ T cells, indicating novel roles for SETD2 in T cell development and lymphomagenesis. Our data render the most comprehensive genetic portrait yet of this uncommon but lethal disease and may inform future classification schemes.

In this paper, we establish some optimality properties of the multiple testing rule induced by the horseshoe estimator due to Carvalho, Polson, and Scott (2010, 2009) from a Bayesian decision theoretic viewpoint. We consider the two-groups model for the data and an additive loss structure such that the total loss is equal to the number of misclassified hypotheses. We use the same asymptotic framework as Bogdan, Chakrabarti, Frommlet, and Ghosh (2011) who introduced the Bayes oracle in the context of multiple testing and provided conditions under which the Benjamini-Hochberg and Bonferroni procedures attain the risk of the Bayes oracle. We prove a similar result for the horseshoe decision rule up to O(1) with the constant in the horseshoe risk close to the constant in the oracle. We use the Full Bayes estimate of the tuning parameter τ. It is worth noting that the Full Bayes estimate cannot be replaced by the Empirical Bayes estimate, which tends to be too small.

Hearing thresholds and wave amplitudes measured using auditory brainstem responses (ABRs) to brief sounds are the predominantly used clinical measures to objectively assess auditory function. However, frequency-following responses (FFRs) to tonal carriers and to the modulation envelope (envelope-following responses or EFRs) to longer and spectro-temporally modulated stimuli are rapidly gaining prominence as a measure of complex sound processing in the brainstem and midbrain. In spite of numerous studies reporting changes in hearing thresholds, ABR wave amplitudes, and the FFRs and EFRs under neurodegenerative conditions, including aging, the relationships between these metrics are not clearly understood. In this study, the relationships between ABR thresholds, ABR wave amplitudes, and EFRs are explored in a rodent model of aging. ABRs to broadband click stimuli and EFRs to sinusoidally amplitude-modulated noise carriers were measured in young (3–6 months) and aged (22–25 months) Fischer-344 rats. ABR thresholds and amplitudes of the different waves as well as phase-locking amplitudes of EFRs were calculated. Age-related differences were observed in all these measures, primarily as increases in ABR thresholds and decreases in ABR wave amplitudes and EFR phase-locking capacity. There were no observed correlations between the ABR thresholds and the ABR wave amplitudes. Significant correlations between the EFR amplitudes and ABR wave amplitudes were observed across a range of modulation frequencies in the young. However, no such significant correlations were found in the aged. The aged click ABR amplitudes were found to be lower than would be predicted using a linear regression model of the young, suggesting altered gain mechanisms in the relationship between ABRs and FFRs with age. These results suggest that ABR thresholds, ABR wave amplitudes, and EFRs measure complementary aspects of overlapping neurophysiological processes and the relationships between these measurements changes asymmetrically with age. Hence, measuring all three metrics provides a more complete assessment of auditory function, especially under pathological conditions like aging.

We provide a framework for assessing the default nature of a prior distribution using the property of regular variation, which we study for global-local shrinkage priors. In particular, we show that the horseshoe priors, originally designed to handle sparsity, are regularly varying and thus are appropriate for default Bayesian analysis. To illustrate our methodology, we discuss four problems of noninformative priors that have been shown to be highly informative for nonlinear functions. In each case, we show that global-local horseshoe priors perform as required. Global-local shrinkage priors can separate a low-dimensional signal from high-dimensional noise even for nonlinear functions.