We consider a sparse linear regression model with unknown symmetric error under the high-dimensional setting. The true error distribution is assumed to belong to the locally \(\beta \)-Hölder class with an exponentially decreasing tail, which does not need to be sub-Gaussian. We obtain posterior convergence rates of the regression coefficient and the error density, which are nearly optimal and adaptive to the unknown sparsity level. Furthermore, we derive the semi-parametric Bernstein-von Mises (BvM) theorem to characterize asymptotic shape of the marginal posterior for regression coefficients. Under the sub-Gaussianity assumption on the true score function, strong model selection consistency for regression coefficients are also obtained, which eventually asserts the frequentist’s validity of credible sets.

我们考虑在高维设置下具有未知对称误差的稀疏线性回归模型。假定真正的错误分布属于局部\（\ beta \）- Hölder类，其尾部呈指数递减，并且不需要是次高斯的。我们获得回归系数和误差密度的后验收敛率，这些收敛率几乎是最佳的并且适应于未知的稀疏度水平。此外，我们推导了半参数Bernstein-von Mises（BvM）定理，以表征回归系数的边际后验的渐近形状。在真实得分函数的亚高斯假设下，还获得了回归系数的强模型选择一致性，从而最终证明了频繁主义者对可信集的有效性。