Bayes Empirical Estimation by the Method of Sieves with some Applications

Let X be a random variable with the density function f(x|γ). Assume that the parameter γ is also a random variable with unknown a priori density θ(γ). In the paper the problem of approximation of a priori density via sieve is considered. The estimate obtained is further used to approximation of Bayes estimator of γ and Bayes risk by some Bayes empirical estimator and Bayes empirical risk. A theorem on almost surely convergence of the Bayes empirical sequence of estimators and Bayes empirical sequence of risks is presented.