We study posterior contraction rates in sparse high-dimensional generalized linear models using priors incorporating sparsity. A mixture of a point mass at zero and a continuous distribution is used as the prior distribution on regression coefficients. In addition to the usual posterior, the fractional posterior, which is obtained by applying the Bayes theorem on a fractional power of the likelihood, is also considered. The latter allows uniformity in posterior contraction over a 15 larger subset of the parameter space. In our setup, the link function of the generalized linear model need not be canonical. We show that Bayesian methods achieve convergence properties analogous to lasso-type procedures. Our results can be used to derive posterior contraction rates in many generalized linear models including logistic, Poisson regression, and others.

中文翻译：

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稀疏广义线性模型中的后收缩

我们使用结合稀疏性的先验来研究稀疏高维广义线性模型中的后收缩率。零点质量和连续分布的混合用作回归系数的先验分布。除了通常的后验之外，还考虑了通过将贝叶斯定理应用于似然的分数幂而获得的分数后验。后者允许在参数空间的 15 个较大子集上实现后收缩的均匀性。在我们的设置中，广义线性模型的链接函数不需要是规范的。我们表明贝叶斯方法实现了类似于套索类型程序的收敛特性。我们的结果可用于在许多广义线性模型中推导出后收缩率，包括逻辑、泊松回归等。