A Mixture Approach to Bayesian Goodness of Fity

We consider a Bayesian approach to goodness of fit, that is, to the problem of testing whether or not a given parametric model is compatible with the data at hand. We thus consider a parametric family F = fF ; 2 g ; where F denotes a cumulative distribution function with parameter . The null hypothesis is H0 : X F for an unknown , that is, there exists such that F (X) U(0; 1). If H0 does not hold, F (X) is a random variable on (0; 1) which is not distributed as U(0; 1). The alternative nonparametric hypothesis can thus be interpreted as F (X) being distributed from a general cdf G on (0; 1), where is infinite dimensional. Instead of using a functional basis as in Verdinelli and Wasserman (1998), we represent G as the (infinite) mixture of Beta distributions, p0U(0; 1) + (1 p0)X k 1 pkBe( k; k) :