  Modern regression applications can involve hundreds or thousands of variables
which motivates the use of variable selection methods. Bayesian variable
selection defines a posterior distribution on the possible subsets of the
variables (which are usually termed models) to express uncertainty about which
variables are strongly linked to the response. This can be used to provide
Bayesian model averaged predictions or inference, and to understand the
relative importance of different variables. However, there has been little work
on meaningful representations of this uncertainty beyond first order summaries.
We introduce Cartesian credible sets to address this gap. The elements of these
sets are formed by concatenating sub-models defined on each block of a
partition of the variables. Investigating these sub-models allow us to
understand whether the models in the Cartesian credible set
always/never/sometimes include a particular variable or group of variables and
provide a useful summary of model uncertainty. We introduce methods to find
these sets that emphasize ease of understanding. The potential of the method is
illustrated on regression problems with both small and large numbers of
variables.
