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