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Say I have 3 random variables ${X, Y, Z}$, and I have collected an iid sample of size $N$ from them: ${\cal D} = \{ (x_i, y_i, z_i), i = 1,\dots,N \}$. The conditional expectation $E[ X | y ]$ can also be written as

\begin{eqnarray*} E[ X | y ] = \int E[ X | y, z ] \,\, dF_{ Z | Y }(z) \end{eqnarray*}

My objective is to learn $E[ X | y ]$ from ${\cal D}$, and my question is: For finite sample size, should I expect any numerical advantage from learning $ E[ X | y, z ], dF_{ Z | Y }(z)$ first, and then integrating out the dependence on z? Or, can I discard the data collected about $z$, and directly learn $E[ X | y ]$.

To make things simpler, I'm ready to assume that my learning strategy is consistent.

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  • $\begingroup$ I do not see how adding a dimension could be of any help. $dF_{Z|Y}$ and $E[X|y,z]$ appear as difficult as $E[X|y]$, if not more so. $\endgroup$
    – ThePawn
    Jan 7, 2013 at 10:25
  • $\begingroup$ Ok, my question was not about the difficulty of carrying these tasks...So, for definiteness, say $X,Y,Z$ are jointly gaussian (thought the parameters of thhis distribution are unknown). Now I can compute estimates for all $E[X|y], E[X|y,z], dF_{Z|Y}(z)$. These estimates will be noisy, as they come from a finite sample size. My question then is: will using $\{z_i\}$ improve my estimates? or, can I discard $\{z_i\}$ from the onset? $\endgroup$
    – Jake
    Jan 7, 2013 at 10:46

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