This paper introduces Dirichlet process mixtures of block $g$ priors for model selection and prediction in linear models. These priors are extensions of traditional mixtures of $g$ priors that allow for differential shrinkage for various (data-selected) blocks of parameters while fully accounting for the predictors' correlation structure, providing a bridge between the literatures on model selection and continuous shrinkage priors. We show that Dirichlet process mixtures of block $g$ priors are consistent in various senses and, in particular, that they avoid the conditional Lindley ``paradox'' highlighted by Som et al.(2016). Further, we develop a Markov chain Monte Carlo algorithm for posterior inference that requires only minimal ad-hoc tuning. Finally, we investigate the empirical performance of the prior in various real and simulated datasets. In the presence of a small number of very large effects, Dirichlet process mixtures of block $g$ priors lead to higher power for detecting smaller but significant effects without only a minimal increase in the number of false discoveries.

翻译：本文针对线性模型中的模型选择与预测问题，提出了基于块$g$先验的狄利克雷过程混合模型。该先验是传统$g$先验混合模型的扩展，允许对（数据选择的）不同参数块进行差异化收缩，同时充分考虑预测变量的相关结构，从而在模型选择文献与连续收缩先验文献之间建立了桥梁。我们证明了基于块$g$先验的狄利克雷过程混合模型在多种意义上具有一致性，特别地，该模型避免了Som等人（2016）强调的条件Lindley“悖论”。此外，我们开发了一种仅需最少人工调参的马尔可夫链蒙特卡罗算法用于后验推断。最后，我们在多种真实与模拟数据集中检验了该先验的实证性能。当存在少量极大效应时，基于块$g$先验的狄利克雷过程混合模型能够在仅轻微增加错误发现数量的前提下，显著提升检测较小但显著效应的统计功效。