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This is page 266 of the Causal Inference book by Rubin/Imbens.

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Note that the argument of the expectation is written down below $E_w$ but not in the first set of equations on the page. IMHO, it makes it much harder to understand the notation at the top of the page rather than the place where it is written down. Is there a reason why the argument of the expectation is omitted so often?

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    $\begingroup$ Yes: repeating the argument every time is useful for beginners, but it becomes cumbersome to write and cumbersome to read. One you reach page 266 of any book, it's unlikely the authors will view their audience as neophytes any longer. Most authors rely on a notational convention to help clarify which variables are random. Some rely on having stated previously which are the random variables. $\endgroup$
    – whuber
    Sep 21, 2021 at 13:02
  • $\begingroup$ But each expectation is different. Its not like you're using one function you defined somewhere, everywhere. $\endgroup$
    – piedpiper
    Sep 21, 2021 at 20:21
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    $\begingroup$ I don't understand. When an author tells you they use bold uppercase latin symbols for random variables, then define "$X$" as a random variable, and later write something like "$E[f(X)],$" where is the ambiguity? It gets a bit more challenging with multiple random variables, as in "$E[E[X\mid Y]]$" (of which the text snippet you provide gives an example), but in such cases a lexical analysis alone will disambiguate the meaning. $\endgroup$
    – whuber
    Sep 21, 2021 at 20:56

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Notation is a tricky thing to do well and consistently. It is pretty rare in a book on any difficult stats topic that the notation is going to be perfect and consistent for the entire book. There is always a tradeoff between being precise (at the expensive of things looking complicated) and concise (at the expensive of not being precise).

For expectation $E[g(X,Y)]$ means take expectation of whatever is random inside the square brackets, i.e. in $g(X,Y)$. If it is clear that (a) both $X$ and $Y$ are random or (b) $X$ is random and $Y$ is fixed or (c) $X$ is fixed and $Y$ is random, then no subscripts are necessary. However, when it is not clear, for example if $X$ is random, but $Y$ is not then it is best to have a subscript as $E_X[g(X,Y)]$.

As it can become difficult to see what is random when using double expectation (as you have in the question), it is helpful to include a subscript. Options include

$E_Y[E_X(X|Y)]$

$E_Y[E(X|Y)]$

I personally find it easy enough to see that the inner expectation is with respect to $X$, so I prefer the second one, but if you want to get really picky, the inner expectation is actually with respect to $X$ conditional on $Y$ so you might write this as

$E_Y[E_{X|Y}(X|Y)]$

but here is where we tip the balance to plenty precise, but not very concise.

Personally I do not find the notation that has a subscript on the inner expectation, but not on the outer helpful.

$E[E_{X}(X|Y)]$

because I have no problem seeing the inner expectation is with respect to $X$. For your notation question in the post it is a question of if the inner expectation does or does not have the subscript, so I do not find that it actually helps to have it.

Too much? This is my first post.

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  • $\begingroup$ I like your answer, thank you. I gain from it. I still think there is an argument to be made for the label to be in both places (inner + outer), but there is much to be said I guess. $\endgroup$
    – piedpiper
    Sep 23, 2021 at 4:42
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    $\begingroup$ Making sense of math symbols is really difficult, so anything that makes it easier for you is what you should try to do when you can. But if you tried to work out proofs you would quickly find that using the more detailed notation becomes tedious. $\endgroup$ Sep 23, 2021 at 19:28

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