We consider Bayesian logistic regression models with group‐structured covariates. In high‐dimensional settings, it is often assumed that only a small portion of groups are significant, and thus, consistent group selection is of significant importance. While consistent frequentist group selection methods have been proposed, theoretical properties of Bayesian group selection methods for logistic regression models have not been investigated yet. In this paper, we consider a hierarchical group spike and slab prior for logistic regression models in high‐dimensional settings. Under mild conditions, we establish strong group selection consistency of the induced posterior, which is the first theoretical result in the Bayesian literature. Through simulation studies, we demonstrate that the proposed method outperforms existing state‐of‐the‐art methods in various settings. We further apply our method to a magnetic resonance imaging data set for predicting Parkinson's disease and show its benefits over other contenders.

Parkinson’s disease is a progressive, chronic, and neurodegenerative disorder that is primarily diagnosed by clinical examinations and magnetic resonance imaging (MRI). In this paper, we propose a Bayesian model to predict Parkinson’s disease employing a functional MRI (fMRI) based radiomics approach. We consider a spike and slab prior for variable selection in high-dimensional logistic regression models, and present an approximate Gibbs sampler by replacing a logistic distribution with a t-distribution. Under mild conditions, we establish model selection consistency of the induced posterior and illustrate the performance of the proposed method outperforms existing state-of-the-art methods through simulation studies. In fMRI analysis, 6216 whole-brain functional connectivity features are extracted for 50 healthy controls along with 70 Parkinson’s disease patients. We apply our method to the resulting dataset and further show its benefits with a higher average prediction accuracy of 0.83 compared to other contenders based on 10 random splits. The model fitting procedure also reveals the most discriminative brain regions for Parkinson’s disease. These findings demonstrate that the proposed Bayesian variable selection method has the potential to support radiological diagnosis for patients with Parkinson’s disease.

High-dimensional variable selection is an important research topic in modern statistics. While methods using nonlocal priors have been thoroughly studied for variable selection in linear regression, the crucial high-dimensional model selection properties for nonlocal priors in generalized linear models have not been investigated. In this paper, we consider a hierarchical generalized linear regression model with the product moment nonlocal prior over coefficients and examine its properties. Under standard regularity assumptions, we establish strong model selection consistency in a high-dimensional setting, where the number of covariates is allowed to increase at a sub-exponential rate with the sample size. The Laplace approximation is implemented for computing the posterior probabilities and the shotgun stochastic search procedure is suggested for exploring the posterior space. The proposed method is validated through simulation studies and illustrated by a real data example on functional activity analysis in fMRI study for predicting Parkinson’s disease.

Variable selection methods with nonlocal priors have been widely studied in linear regression models, and their theoretical and empirical performances have been reported. However, the crucial model selection properties for hierarchical nonlocal priors in high-dimensional generalized linear regression have rarely been investigated. In this paper, we consider a hierarchical nonlocal prior for high-dimensional logistic regression models and investigate theoretical properties of the posterior distribution. Specifically, a product moment (pMOM) nonlocal prior is imposed over the regression coefficients with an Inverse-Gamma prior on the tuning parameter. Under standard regularity assumptions, we establish strong model selection consistency in a high-dimensional setting, where the number of covariates is allowed to increase at a sub-exponential rate with the sample size. We implement the Laplace approximation for computing the posterior probabilities, and a modified shotgun stochastic search procedure is suggested for efficiently exploring the model space. We demonstrate the validity of the proposed method through simulation studies and an RNA-sequencing dataset for stratifying disease risk.