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.

翻译：本文介绍了两个尚未解决的贝叶斯估算问题的最佳结果:对平方总变差函数的抽样分布估计和对1美元平方损失函数密度的估计。后端预测分布为这些问题提供了解决办法,举出了一些例子加以说明。还述及了贝叶斯分布函数的估计问题。证明了密度估计值的一致性。