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Usually, what I read is taking conditional expecation with respect to X or Z, a single random variable, but in the book "elements of statistical learning" page 291,given $T=(Z,Z^m)$ and $Z^m$ is missing data:

we have

$l(Q';Z)=l_0(Q';T)-l_1(Q';Z^m|Z)$

where $l_1$ is based on the conditional density $Pr(Z^m|Z,Q')$.

Taking conditional expectations with respect to the distribution of T|Z governed by parameter Q gives:

$l(Q';Z)=E[l_0(Q';T)|Z,Q]-E[l_1(Q';Z^m|Z)|Z,Q]$

I don't quite get the idea of distribution of $T|Z$, and how come the likelihood function became the Expectation form. Thanks!

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The distribution of $T|Z$ is exactly what it sounds like.. It's the distribution of all of the data conditioned on the values of the non-missing data. All they are doing in that equation is taking the conditional expected value of both sides of the equation with respect to $Z$ (and under parameter $Q$). They took the expected value of both sides because we don't know the values of $Z^m$ so we need to average over them. The left-hand side $l(Q',Z)$ only depends on the data through $Z$ so it's not affected by the conditional expectation wrt $Z$ (thus we appear to get more than our money's worth from the decision to take this expectation, since the goal is to calculate $l(Q',Z)$).

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    thanks for your kind reply! then as for the expectation, if I wrote the formula into integration form, should I use $dz$ or $dt$, that's the place I'm confused about – 2017-02-07
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    @Griffan you'll have to explain more what you mean by put the formula into 'integration form'. – 2017-02-07
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    I mean $E(x)=\int{x*p(x)}dx$, if I want to expand the above expectation formula, what is the $dx$ here, $dt$ or $dz$? @spaceisdarkgreen – 2017-02-07
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    Sorry, I'm having a hard time articulating this. You are taking a conditional expectation wrt $Z.$ As a simple example say $T =(Z,Z_m)$ and $Z$ and $Z_m$ are two correlated normal variables or something. And you want to take the conditional expectation of some function $f(Z,Z_m)$ wrt $Z.$ You would take $\int f(z,z_m)p_{Z,Z_m|Z}(z,z_m|Z) dz dz_m = \int f(Z,z_m) p_{Z|Z_m}(z_m|Z) dz_m $ – 2017-02-07