In the problem of regions, we wish to know which one of a discrete
set of possibilities applies to a continuous parameter vector. This problem
arises in the following way: we compute a descriptive statistic from a set of
data, notice an interesting feature and wish to assign a confidence level to
that feature. For example, we compute a density estimate and notice that the
estimate is bimodal. What confidence can we assign to bimodality? A natural way
to measure confidence is via the bootstrap: we compute our descriptive
statistic on a large number of bootstrap data sets and record the proportion of
times that the feature appears. This seems like a plausible measure of
confidence for the feature. The paper studies the construction of such
confidence values and examines to what extent they approximate frequentist
$p$-values and Bayesian a posteriori probabilities. We derive more
accurate confidence levels using both frequentist and objective Bayesian
approaches. The methods are illustrated with a number of examples, including
polynomial model selection and estimating the number of modes of a density.