  Optimality results for two outstanding Bayesian estimation problems are given
in this paper: the estimation of the sampling distribution for the squared
total variation function and the estimation of the density for the
$L^1$-squared loss function. The posterior predictive distribution provides the
solution to these problems. Some examples are presented to illustrate it. The
Bayesian estimation problem of a distribution function is also addressed.
Consistency of the estimator of the density is proved.
