On Bayes Procedures

Abstract

In this chapter we describe some of the asymptotic properties of Bayes procedures. These are obtained by using on the parameter set Θ a finite positive measure μ and minimizing the average risk $\int R(\theta,\rho)\mu(d\theta)$. (See Chapter 2 for notation). The procedure ρ that achieves this minimum will of course depend on the choice of μ. However the literature contains numerous statements to the effect that, for large samples, the choice of μ matters little. This cannot be generally true, but we start with a proposition to this effect. If instead of μ one uses A dominated by μ and if the density $\frac{d\lambda}{d\mu}$ can be closely estimated, then a procedure that is nearly Bayes for μ is also nearly Bayes for λ.