We study full Bayesian procedures for sparse linear regression when errors have a symmetric but otherwise unknown distribution. The unknown error distribution is endowed with a symmetrized Dirichlet process mixture of Gaussians. For the prior on regression coefficients, a mixture of point masses at zero and continuous distributions is considered. We study behavior of the posterior with diverging number of predictors. Conditions are provided for consistency in the mean Hellinger distance. The compatibility and restricted eigenvalue conditions yield the minimax convergence rate of the regression coefficients in ℓ 1 -and ℓ 2 -norms, respectively. The convergence rate is adaptive to both the unknown sparsity level and the unknown symmetric error density under compatibility conditions. In addition, strong model selection consistency and a semi-parametric Bernstein-von Mises theorem are proven under slightly stronger conditions.Keywords: Adaptive contraction rates, Bernstein von-Mises theorem, Dirichlet process mixture, high-dimensional semiparametric model, sparse prior, symmetric error * This article has been accepted for publication in Information and Inference Published by Oxford University Press. The accepted version contains significantly improved results, which is available at https://doi.org/10.1093/imaiai/iay022 the high-dimensional setting where p, the number of the predictors and the size of the coefficient vector, may grow with the sample size n, and possibly p ≫ n. If p > n, model (1) is not identifiable due to the singularity of its design matrix, therefore θ is not estimable unless further restrictions or structures are imposed. A standard assumption for θ is the sparsity condition which assumes that most components of θ are zero. For the last two decades, model (1) has been extensively studied under various sparsity conditions, in particular through penalized regression approaches such as Lasso and its various variants or extensions [36,37,46,47]. Recent advances in MCMC and other computational algorithms have led to a growing development of Bayesian models incorporating sparse priors [7,8,13,20,27]. In general, two classes of sparse priors are often used, the first being the spike and slab type (see e.g., [7,8]), with some recent work extending to continuous versions [20,28,32,33], and the other being continuous shrinkage priors; in particular, local-global shrinkage priors (see [1,5,31]).In the literature, both frequentist and Bayesian, the standard Gaussian error model, in which ǫ i 's are assumed to be i.i.d. Gaussian, is typically adopted, providing substantial computational and theoretical benefits. Using a squared error loss function, various penalization techniques are developed. Theoretical aspects of such estimates have been explored, showing recovery of θ in nearly optimal rate or optimal selection of the true nonzero coefficients [3,7,11,12,21]. More recent theoretical advances assure that relying on certain desparsifying techniques, asymptotically optimal (or at least honest) confidence set...

In a smooth semi-parametric model, the marginal posterior distribution for a finite dimensional parameter of interest is expected to be asymptotically equivalent to the sampling distribution of any efficient point-estimator. The assertion leads to asymptotic equivalence of credible and confidence sets for the parameter of interest and is known as the semi-parametric Bernstein-von Mises theorem. In recent years, it has received much attention and has been applied in many examples. We consider models in which errors with symmetric densities play a role; more specifically, it is shown that the marginal posterior distributions of regression coefficients in the linear regression and linear mixed effect models satisfy the semi-parametric Bernstein-von Mises assertion. As a consequence, Bayes estimators in these models achieve frequentist inferential optimality, as expressed e.g. through Hájek's convolution and asymptotic minimax theorems.For the prior on the space of error densities, we provide two well-known examples, the Dirichlet process mixture of normal densities and random series priors.Particularly, the result provides an efficient estimate of regression coefficients in the linear mixed effect model, for which no other efficient point-estimator was known previously.

Recent advances in the personality and turnover literatures suggest the importance of expanding current turnover criteria, incorporating dark personality traits, and examining the role of time in these relationships. The present study investigates these issues by considering both the speed and the reasons for leaving, examining a wider range of personality variables as predictors by including both “bright” and “dark” traits, and exploring the potential moderating effect of time in such predictions. Data were collected from a sample of 617 employees working in an electronics manufacturing firm in the United States. Using a Bayesian survival analysis framework, we found that dark traits were just as useful in predicting turnover outcomes as traditional personality traits and best predicted the specific turnover reasons, “deviant behavior” and “no call no show.” Investigating the role of time showed that job satisfaction and intellectual curiosity (i.e., Openness) grew in predictive strength over the course of organizational tenure but that the time-dependent effects of other predictors were negligible.

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