On Bayesian estimation of densities and sampling distributions: The posterior predictive distribution as the Bayes estimator

Optimality results for three interesting Bayesian estimation problems are presented in this paper: the estimation of the sampling distribution for the squared total variation function, the estimation of the density for the $L^1$-squared loss function and the estimation of a real distribution function for the $L^{\infty }$-squared loss function. The posterior predictive distribution provides the solution to these problems. Some examples are presented to illustrate it.