An alternative to the standard Bayesian procedure for discrimination between normal linear models

We consider the standard Bayesian procedure for discrimination, focusing on its tendency to give low posterior probabilities to some correct models, even in conditions of moderate to large sampling sizes. Furthermore, in some situations that could well occur in practice, the standard procedure is even asymptotically inconsistent. We claim that the explanation is that the standard procedure inflates the posterior probability of the model that has the smallest expected increase in information about its parameters. We then propose to modify it by an amount related to the expected gain in information, and show that the ‘Lindley paradox’ does not occur. Finally, we obtain the limiting alternative criterion that results by letting the prior densities become noninformative, which enables us to measure how sensitive the discrimation is to the informative prior densities used.