Resumen
In a normal example of Stone (1979, J. Roy. Statist. Soc. Ser. B 41, 276–278), Berger et al. (2003, J. Statist. Plann. Inference 112, 241–258) showed BIC may be a poor approximation to the logarithm of Bayes Factor. They proposed a Generalized Bayes Information Criterion (GBIC) and a Laplace approximation to the log Bayes factor in that problem. We consider a fairly general case where one has p groups of observations coming from an arbitrary general exponential family with each group having a different parameter and r observations. We derive a GBIC and a Laplace approximation to the integrated likelihood, under the assumption that p?8 and r?8 (and some additional restrictions, which vary from example to example). The general derivation clarifies the structure of GBIC. A general theorem is presented to prove the accuracy of approximation, and the worst possible approximation error is derived for several examples. In several numerical examples, the Laplace approximation and GBIC are seen to be quite good. They perform much better than BIC.

Resumen
We establish weak and strong posterior consistency of Gaussian process priors studied by Lenk [1988. The logistic normal distribution for Bayesian, nonparametric, predictive densities. J. Amer. Statist. Assoc. 83 (402), 509-516] for density estimation. Weak consistency is related to the support of a Gaussian process in the sup-norm topology which is explicitly identified for many covariance kernels. In fact we show that this support is the space of all continuous functions when the usual covariance kernels are chosen and an appropriate prior is used on the smoothing parameters of the covariance kernel. We then show that a large class of Gaussian process priors achieve weak as well as strong posterior consistency (under some regularity conditions) at true densities that are either continuous or piecewise continuous.