A new Bayesian information criterion for the linear regression model: case of maximum a posteriori estimator

This article is concerned with the Bayesian selection of linear regression models. In order to achieve this objective, we will successively present the estimators obtained by maximum a posteriori in the two cases of a priori laws of the parameters 𝜷 and 𝝈 𝟐 : cases informative and non-informative one. In the non-informative case, we put forth the use of Jeffreys’ a priori law, which is based on Fisher’s information. In the informative case, we examine the a priori joint distribution of these two parameters, which follow a normal-gamma distribution. We then present some properties of these estimators for both cases. Based on these properties, we propose a new model selection criterion based on the calculation of the Kullback-Leibler divergence between the operational model (approximate) and the true model (unknown)