The median probability model (MPM) Barbieri and Berger (2004) is defined as
the model consisting of those variables whose marginal posterior probability of
inclusion is at least 0.5. The MPM rule yields the best single model for
prediction in orthogonal and nested correlated designs. This result was
originally conceived under a specific class of priors, such as the point mass
mixtures of non-informative and g-type priors. The MPM rule, however, has
become so very popular that it is now being deployed for a wider variety of
priors and under correlated designs, where the properties of MPM are not yet
completely understood. The main thrust of this work is to shed light on
properties of MPM in these contexts by (a) characterizing situations when MPM
is still safe under correlated designs, (b) providing significant
generalizations of MPM to a broader class of priors (such as continuous
spike-and-slab priors). We also provide new supporting evidence for the
suitability of g-priors, as opposed to independent product priors, using new
predictive matching arguments. Furthermore, we emphasize the importance of
prior model probabilities and highlight the merits of non-uniform prior
probability assignments using the notion of model aggregates
