  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.
