On high-dimensional classification by sparse generalized Bayesian logistic regression

Abstract This work addresses the problem of high-dimensional classification by exploring a generalized Bayesian logistic regression method under sparsity-inducing prior distributions. The method involves utilizing a fractional power of the likelihood, resulting in the fractional posterior. Our study yields concentration results for the fractional posterior, not only on the joint distribution of the predictors and response variable but also for the regression coefficients. Subsequently, we derive novel findings concerning misclassification excess risk bounds using sparse generalized Bayesian logistic regression. Our results showcase the ability of our method to adapt to the unknown sparsity level. The results align with recent findings on penalized methods in the frequentist literature while requiring fewer assumptions. Moreover, we also present an analysis addressing model misspecification. Simulations and real data analyses are undertaken to validate our theoretical results.