论文标题
贝叶斯高维半参数推断超出高斯错误
Bayesian High-dimensional Semi-parametric Inference beyond sub-Gaussian Errors
论文作者
论文摘要
我们考虑一个稀疏的线性回归模型，在高维设置下具有未知的对称误差。假定真正的误差分布属于本地$β$-Hölder类，其尾巴呈指数减小，而这不需要是次高斯。我们获得回归系数和误差密度的后收敛速率，它们几乎最佳且适应未知的稀疏度。此外，我们得出了半参数Bernstein-von Mises（BVM）定理，以表征边缘后部渐近形状的回归系数。在对真实分数函数的次高西度性假设下，还获得了回归系数的强模型选择一致性，这最终主张了频繁的可靠集合的有效性。
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 $β$-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.
