We present simple conditions for Bayesian consistency in the supremum metric. The key to the technique is a triangle inequality which allows us to explicitly use weak convergence, a consequence of the standard Kullback--Leibler support condition for the prior. A further condition is to ensure that smoothed versions of densities are not too far from the original density, thus dealing with densities which could track the data too closely. A key result of the paper is that we demonstrate supremum consistency using weaker conditions compared to those currently used to secure $\mathbb{L}_1$ consistency.
翻译:我们为贝叶斯人达到最高标准的一致性提出了简单的条件。技术的关键在于三角不平等,这种不平等使我们能够明确使用弱的趋同性,这是标准 Kullback-Libel支持条件对前一个条件的后果。另一个条件是,确保平滑的密度不远于原始密度,从而处理过于密切地跟踪数据的密度问题。文件的一个关键结果是,我们使用比目前用来确保$\mathbb{L ⁇ 1$一致性的条件更弱的条件来显示超强的趋同性。