论文标题
在紧凑的空间上存在匹配的先验
Existence of matching priors on compact spaces
论文作者
论文摘要
先验是$ 1-α$的匹配先验，因此相关的$ 1-α$可靠套件也是$ 1-α$置信度。我们研究了可靠地区的一般家庭的匹配先验。我们的主要结果给出了拓扑条件，在这些条件下，存在针对可靠地区特定家族的匹配先验。非正式地，我们证明，在紧凑的参数空间上，如果我们在先验上采用Wasserstein Metric，则存在匹配的先验。鉴于这一总体结果，我们观察到可靠的区域的典型家庭，例如可靠的球，最高的密度区域，分位数等，无法满足这种拓扑状态。我们展示了如何设计符合这些拓扑条件的近似后验可信球和最高后的密度区域，从而产生匹配的先验。最后，我们评估了一种基于离散化和迭代的计算近似匹配先验的数值方案。我们的主要定理的证明使用了非标准分析中的工具，并建立了有关可能具有独立关注的Wasserstein指标扩展的新结果。
A matching prior at level $1-α$ is a prior such that an associated $1-α$ credible set is also a $1-α$ confidence set. We study the existence of matching priors for general families of credible regions. Our main result gives topological conditions under which matching priors for specific families of credible regions exist. Informally, we prove that, on compact parameter spaces, a matching prior exists if the so-called rejection-probability function is jointly continuous when we adopt the Wasserstein metric on priors. In light of this general result, we observe that typical families of credible regions, such as credible balls, highest-posterior density regions, quantiles, etc., fail to meet this topological condition. We show how to design approximate posterior credible balls and highest-posterior-density regions that meet these topological conditions, yielding matching priors. Finally, we evaluate a numerical scheme for computing approximately matching priors based on discretization and iteration. The proof of our main theorem uses tools from nonstandard analysis and establishes new results about the nonstandard extension of the Wasserstein metric that may be of independent interest.
