Posterior Consistency for Coefficient Estimation and Model Selection in the General Linear Hypothesis

Abstract

Berk (1970), LeCam (1953) and others have given conditions for the consistency of posterior distributions from a sequence of random variables. They have required that the sequence be i.i.d. We show that their results, Berk's in particular, may be extended to the general linear hypothesis with normal errors model (where the sequence of observations of the dependent variable need not be i.i.d.). We do not assume that the distribution governing the sequence of dependent variables has a regression function which satisfies the assumed model nor do we assume its errors are normal. Consistency is shown for both fixed and random sampling designs. We show that the convergence is to a projection of only the true regression function upon the space of regression functions given by the model. Finally, we assume that several such models are under consideration, each with a prior probability. We determine conditions for the a.s. convergence of their posterior probabilities to a degenerate distribution. Not all these conditions may be derived by any simple extension of Berk's results.