Approximating priors by finite mixtures of conjugate distributions for an exponential family

The problem of approximating a prior by a suitable finite mixture of distributions, with respect to Prokhorov’s metric, is considered. We give explicit estimates of the error of approximation when the statistical model is a suitable exponential family. An illustrative example shows an application of the approximation technique. In an attempt to generalize these results to an arbitrary natural exponential family model, rates of convergence of the mixture to the approximated prior are given.