We propose a family of priors that satisfies the second-order probability matching property. The posterior quantile of a probability matching prior is exactly or approximately equal to the frequentist one. Most models lack an exact matching prior. If all quantiles of a prior’s posterior converge to the frequentist ones up to $$o(n^{-1/2})$$o(n-1/2) or $$o(n^{-1})$$o(n-1) as the sample size n increases, the prior is called a first-order probability matching prior and a second-order probability matching prior, respectively. Although a second-order matching prior does not necessarily exist, a first-order matching prior always exists. We introduce a class of priors that depend on the sample size and matching probability. We derive the condition under which the family satisfies the second-order probability matching property even when a second-order probability matching prior does not exist. The superiority of the proposed priors is illustrated in several numerical experiments.

中文翻译：

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以样本大小和匹配概率为参数的二阶匹配先验族

我们提出了一系列满足二阶概率匹配属性的先验。概率匹配先验的后分位数恰好或近似等于频率主义者的分位数。大多数模型缺乏精确匹配的先验。如果先验后验的所有分位数都收敛到频率派的分位数达到 $$o(n^{-1/2})$$o(n-1/2) 或 $$o(n^{-1})$ $o(n-1) 随着样本大小 n 的增加，先验分别称为一阶概率匹配先验和二阶概率匹配先验。虽然二阶匹配先验不一定存在，但一阶匹配先验总是存在的。我们引入了一类取决于样本大小和匹配概率的先验。即使二阶概率匹配先验不存在，我们也推导出了家庭满足二阶概率匹配属性的条件。几个数值实验说明了所提出的先验的优越性。