Zellner's $g$‐prior is one of the most popular choices for model selection in Bayesian linear regression. Despite its popularity, the asymptotic theory for high‐dimensional variable selection is not yet fully developed. In this paper, we investigate the asymptotic behaviour of Bayesian model selection under the $g$‐prior as the model dimension grows with the sample size. We find a simple and intuitive condition under which the posterior model distribution tends to be concentrated on the true model as the sample size increases even if the number of predictors grows much faster than the sample size does. Simulation study results indicate that satisfaction of this condition is essential for the success of Bayesian high‐dimensional variable selection under the $g$‐prior.

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关于默认先验条件下贝叶斯高维变量选择的一致性的注释

泽尔纳的 $G$在贝叶斯线性回归中，先验模型选择是最受欢迎的选择之一。尽管它很流行，但高维变量选择的渐近理论尚未完全发展。在本文中，我们研究了贝叶斯模型选择在条件下的渐近行为。$G$随模型尺寸的增加而增加。我们发现一个简单而直观的条件，即在后验模型分布趋于集中于真实模型的情况下，即使样本量的增长速度远快于样本量，样本量也会增加。仿真研究结果表明，该条件的满足对于贝叶斯高维变量选择成功的关键。$G$以前。