We study high-dimensional Bayesian linear regression with a general beta prime distribution for the scale parameter. Under the assumption of sparsity, we show that appropriate selection of the hyperparameters in the beta prime prior leads to the (near) minimax posterior contraction rate when $p \gg n$. For finite samples, we propose a data-adaptive method for estimating the hyperparameters based on marginal maximum likelihood (MML). This enables our prior to adapt to both sparse and dense settings, and under our proposed empirical Bayes procedure, the MML estimates are never at risk of collapsing to zero. We derive efficient Monte Carlo EM and variational EM algorithms for implementing our model, which are available in the R package NormalBetaPrime. Simulations and analysis of a gene expression data set illustrate our model's self-adaptivity to varying levels of sparsity and signal strengths.

中文翻译：

---

关于高维贝叶斯回归模型中尺度参数的 Beta Prime 先验

我们研究了高维贝叶斯线性回归，其尺度参数具有一般的 beta 素数分布。在稀疏性的假设下，我们表明，当$ p \ gg n $时，我们认为在测试中的近似参数的近似数目的近额表明导致（近）最低收缩率。对于有限样本，我们提出了一种基于边际最大似然（MML）估计超参数的数据自适应方法。这使我们的先验能够同时适应稀疏和密集设置，并且在我们提出的经验贝叶斯程序下，MML 估计永远不会有崩溃为零的风险。我们推导出了高效的 Monte Carlo EM 和变分 EM 算法来实现我们的模型，这些算法在 R 包 NormalBetaPrime 中可用。基因表达数据集的模拟和分析说明了我们的模型