We develop a framework to study posterior contraction rates in sparse high dimensional generalized linear models (GLM). We introduce a new family of GLMs, denoted by clipped GLM, which subsumes many standard GLMs and makes minor modification of the rest. With a sparsity inducing prior on the regression coefficients, we delineate sufficient conditions on true data generating density that leads to minimax optimal rates of posterior contraction of the coefficients in $\ell_1$ norm. Our key contribution is to develop sufficient conditions commensurate with the geometry of the clipped GLM family, propose prior distributions which do not require any knowledge of the true parameters and avoid any assumption on the growth rate of the true coefficient vector.

翻译：我们开发了一个框架来研究稀疏高维通用线性模型的后继收缩率(GLM ) 。 我们引入了一个新的GLM系列,由剪切的GLM 表示,它包含许多标准GLM,并对其余部分稍作修改。随着在回归系数之前的松散诱导,我们为产生密度的真实数据划定了充分的条件,从而导致以$@ell_1美元标准计算系数后继收缩率的最小最大最佳比率。我们的主要贡献是开发与剪切切除的GLM 家族的几何学相称的充足条件,提出不需要对真实参数有任何了解的先前分配,避免对真实系数矢量增长率的任何假设。