Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A General Derivative Identity for the Conditional Expectation with Focus
  on the Exponential Family   Consider a pair of random vectors $(\mathbf{X},\mathbf{Y}) $ and the
conditional expectation operator
$\mathbb{E}[\mathbf{X}|\mathbf{Y}=\mathbf{y}]$. This work studies analytic
properties of the conditional expectation by characterizing various derivative
identities. The paper consists of two parts. In the first part of the paper, a
general derivative identity for the conditional expectation is derived.
Specifically, for the Markov chain $\mathbf{U} \leftrightarrow \mathbf{X}
\leftrightarrow \mathbf{Y}$, a compact expression for the Jacobian matrix of
$\mathbb{E}[\mathbf{U}|\mathbf{Y}=\mathbf{y}]$ is derived. In the second part
of the paper, the main identity is specialized to the exponential family.
Moreover, via various choices of the random vector $\mathbf{U}$, the new
identity is used to recover and generalize several known identities and derive
some new ones. As a first example, a connection between the Jacobian of $
\mathbb{E}[\mathbf{X}|\mathbf{Y}=\mathbf{y}]$ and the conditional variance is
established. As a second example, a recursive expression between higher order
conditional expectations is found, which is shown to lead to a generalization
of the Tweedy's identity. Finally, as a third example, it is shown that the
$k$-th order derivative of the conditional expectation is proportional to the
$(k+1)$-th order conditional cumulant.

17
The categories of this paper are: q-fin
0
The categories of this paper are: hep-ph
13
The categories of this paper are: quant-ph
7
The categories of this paper are: hep-th
31
The categories of this paper are: ao-sci
28
The categories of this paper are: alg-geom
5
The categories of this paper are: gr-qc
32
The categories of this paper are: atom-ph
23
The categories of this paper are: solv-int
35
The categories of this paper are: mtrl-th
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
19
The categories of this paper are: eess
37
The categories of this paper are: supr-con
25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex