The authors consider the problem of optimal estimation of a random variable X based on an observation denoted by a random vector Y. A commonly encountered problem involves estimating X via h(Y) so as to minimize E( Phi (X-h(Y))), where h is Borel measurable and Phi is a Borel measurable cost function chosen to adequately reflect the fidelity demands of the problem under consideration. The authors place a mild condition on the regular conditional distribution of X given sigma (Y) that ensures that E( Phi (X-h(Y))) is minimized for any cost function Phi that is nonnegative, even and convex. In addition, it is shown that given any Borel measurable function g: R to R, there exist random variables X and Y possessing a joint density function such that E(X mod Y=y)=g(y) almost everywhere with respect to Lebesgue measure.
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机译：作者考虑了基于以随机向量Y表示的观测值对随机变量X进行最佳估计的问题。常见的问题涉及通过h（Y）估计X，以使E（Phi（Xh（Y））最小化。 ，其中h是Borel可测量的成本函数，Phi是Borel可测量的成本函数，选择该函数可充分反映所考虑问题的保真度要求。作者在给定的西格玛（Y）的X的规则条件分布上放置了温和条件，以确保对于任何非负，偶数和凸的成本函数Phi，E（Phi（X-h（Y）））最小。此外，表明给定任何Borel可测量函数g：R到R，存在随机变量X和Y，它们具有联合密度函数，使得相对于E（X mod Y = y）= g（y）几乎处处勒贝格措施。
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