#### Optimal predictive model selection

Maria Maddalena Barbieri and James Berger

Often the goal of model selection is to choose a model for future
prediction, and it is natural to measure the accuracy of a future
prediction by squared error loss. Under the Bayesian approach, it
is commonly perceived that the optimal predictive model is the
model with highest posterior probability, but this is not
necessarily the case. In this paper we show that, for selection
among normal linear models, the optimal predictive model is often
the {\it median probability model}, which is defined as the model
consisting of those variables which have overall posterior
probability greater than or equal to 1/2 of being in a model. The
median probability model often differs from the highest
probability model.