Researchers frequently wish to assess the equality or inequality of groups, but this comes with the challenge of adequately adjusting for multiple comparisons. Statistically, all possible configurations of equality and inequality constraints can be uniquely represented as partitions of the groups, where any number of groups are equal if they are in the same partition. In a Bayesian framework, one can adjust for multiple comparisons by constructing a suitable prior distribution over all possible partitions. Inspired by work on variable selection in regression, we propose a class of flexible beta-binomial priors for Bayesian multiple comparison adjustment. We compare this prior setup to the Dirichlet process prior suggested by Gopalan and Berry (1998) and multiple comparison adjustment methods that do not specify a prior over partitions directly. Our approach to multiple comparison adjustment not only allows researchers to assess all pairwise (in)equalities, but in fact all possible (in)equalities among all groups. As a consequence, the space of possible partitions grows quickly - for ten groups, there are already 115,975 possible partitions - and we set up a stochastic search algorithm to efficiently explore the space. Our method is implemented in the Julia package EqualitySampler, and we illustrate it on examples related to the comparison of means, variances, and proportions.

翻译：研究者常需评估组间均等性或非均等性，但随之而来的是如何充分调整多重比较的挑战。从统计学角度，所有均等与非均等约束的可能配置可唯一表示为组的分区，其中同一分区内的任意组数视为均等。在贝叶斯框架下，可通过构建所有可能分区上的适当先验分布来实现多重比较调整。受回归中变量选择研究的启发，我们提出一类灵活的贝塔-二项先验用于贝叶斯多重比较调整。我们将该先验设置与Gopalan和Berry(1998)提出的狄利克雷过程先验以及未直接指定分区先验的多重比较调整方法进行比较。我们的多重比较调整方法不仅允许研究者评估所有成对(非)均等性，事实上还能评估所有组间的全部可能(非)均等性。因此，可行分区空间增长迅速——对于十个组，已有115,975种可能分区——我们设计了随机搜索算法以高效探索该空间。该方法已在Julia软件包EqualitySampler中实现，并通过均值、方差和比例比较的实例进行展示。