A Mixture Approach to Bayesian Goodness of Fit
Christian P. Robert, Judith Rousseau · 2002
this paper, this assessment is paramount given that we are concerned with a goodness of fit perspective. The quantity of interest is then the distance between the true density and the proposed model, d(f, ). As is often the case in nonparametric inference, we consider the Hellinger distance between two distributions F and G, defined as d(F, G) = # dF # dG . Since we are only concerned with distributions absolutely continuous with respect to Lebesgue measure, we also denote d(f, g) the Hellinger distance between F and G, where f, g are the densities with respect to the Lebesgue measure of F and G respectively. Then we define = inf d(f, f # ) . We approximate this quantity using its posterior expectation E ], for some prior # on (#, #). To test the parametric model, we must therefore compare the above posterior expectation with some reference quantity. Actually, the Bayes estimate under the loss function : L(#, f) = a 0 d(f, a 1 (2 F)) (2) is given by #(X ) = 0, i.e. we accept the null hypothesis, if and only if E 2a 1 /(a 0 + a 1 ). In the general case, the choice of (a 0 , a 1 ) is quite arbitrary. We therefore propose in this paper a way to calibrate 2a 1 /(a 0 + a 1 ). In particular, a 0 should increase as the number of observations becomes larger. The informal perspective on this point is that if the parametric model is not far from the true model, it is better to use such a model, especially when the number of observations is not large. In other words, the smaller the sample size is, the more relevant the parametric model might get. The idea is then to compare E ] with a quantity that would characterize its behaviour under the null hypothesis. A usual way to do it is to use posterior predictive p-values, see for instance M...