Consider a channel ${\\bf Y}={\\bf X}+ {\\bf N}$ where ${\\bf X}$ is an\n$n$-dimensional random vector, and ${\\bf N}$ is a Gaussian vector with a\ncovariance matrix ${\\bf \\mathsf{K}}_{\\bf N}$. The object under consideration in\nthis paper is the conditional mean of ${\\bf X}$ given ${\\bf Y}={\\bf y}$, that\nis ${\\bf y} \\to E[{\\bf X}|{\\bf Y}={\\bf y}]$. Several identities in the\nliterature connect $E[{\\bf X}|{\\bf Y}={\\bf y}]$ to other quantities such as the\nconditional variance, score functions, and higher-order conditional moments.\nThe objective of this paper is to provide a unifying view of these identities.\n In the first part of the paper, a general derivative identity for the\nconditional mean is derived. Specifically, for the Markov chain ${\\bf U}\n\\leftrightarrow {\\bf X} \\leftrightarrow {\\bf Y}$, it is shown that the Jacobian\nof $E[{\\bf U}|{\\bf Y}={\\bf y}]$ is given by ${\\bf \\mathsf{K}}_{{\\bf N}}^{-1}\n{\\bf Cov} ( {\\bf X}, {\\bf U} | {\\bf Y}={\\bf y})$.\n In the second part of the paper, via various choices of ${\\bf U}$, the new\nidentity is used to generalize many of the known identities and derive some new\nones. First, a simple proof of the Hatsel and Nolte identity for the\nconditional variance is shown. Second, a simple proof of the recursive identity\ndue to Jaffer is provided. Third, a new connection between the conditional\ncumulants and the conditional expectation is shown. In particular, it is shown\nthat the $k$-th derivative of $E[X|Y=y]$ is the $(k+1)$-th conditional\ncumulant.\n The third part of the paper considers some applications. In a first\napplication, the power series and the compositional inverse of $E[X|Y=y]$ are\nderived. In a second application, the distribution of the estimator error\n$(X-E[X|Y])$ is derived. In a third application, we construct consistent\nestimators (empirical Bayes estimators) of the conditional cumulants from an\ni.i.d. sequence $Y_1,...,Y_n$.\n
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