The current definition of a Bayesian credible set cannot, in general, achieve an arbitrarily preassigned credible level. This drawback is particularly acute for classification problems, where there are only a finite number of achievable credible levels. As a result, there is as of today no general way to construct an exact credible set for classification. In this paper, we introduce a generalized credible set that can achieve any preassigned credible level. The key insight is a simple connection between the Bayesian highest posterior density credible set and the Neyman--Pearson lemma, which, as far as we know, hasn't been noticed before. Using this connection, we introduce a randomized decision rule to fill the gaps among the discrete credible levels. Accompanying this methodology, we also develop the Steering Wheel Plot to represent the credible set, which is useful in visualizing the uncertainty in classification. By developing the exact credible set for discrete parameters, we make the theory of Bayesian inference more complete.

翻译：当前贝叶斯可信集的定义通常无法达到任意预设的可信水平。这一缺陷在分类问题中尤为突出，因为此时仅存在有限个可达的可信水平。因此，至今尚无通用方法为分类问题构建精确可信集。本文引入一种可达到任意预设可信水平的广义可信集。关键洞见在于贝叶斯最高后验密度可信集与内曼-皮尔逊引理之间尚未被注意到的简单关联。借助这一关联，我们引入随机化决策规则来填补离散可信水平之间的空隙。伴随该方法论，我们还开发了方向盘图来可视化分类中的不确定性。通过为离散参数建立精确可信集，我们进一步完善了贝叶斯推断理论。