Are $E_{Y|X}[Y]$ and $E_{Y|X}[Y|X]$ equivalent?

Question

I am under the impression that:

$E_{Y|X}[Y] = E_{Y|X}[Y|X] = \int y f_{y|X}(y|X) dy$

And by extension:

$f_{y|X}(y) = f_{y|X}(y|X)$

Please correct me if I am wrong. Thank you! I understand that notation like this is dependent on whatever the author intends, so I am mostly hoping to ask for what convention you have all seen most commonly.

Here are some places where I've seen notation like this:

(Here $|X$ is in the subscript only, but the author seems to intend for $E[Y|X=x] = E_{Y|X}[Y]$)

(Here $|X$ is in both the subscript and the argument)

Answer 1

These notations with subscripts are an abomination used only by people who don't know better.

There are conditional probabilitites, and hence conditional probability distributions and one can speak of $\operatorname E(X\mid A)$ where $X$ is a random variable and $A$ is an event.

The event in question may be that a particular random variable has a particular value, and then one can write $\operatorname E(X\mid Y=y),$ where $Y$ is a random variable and $y$ is any of the values it can take.

This conditional expected value, $\operatorname E(X\mid Y=y),$ depends on what number $y$ is, so one may write $\operatorname E(X\mid Y=y) = h(y).$ In that case, one can define $\operatorname E(X\mid Y)=h(Y),$ and that is a random variable in its own right, and is determined by the value of the random variable $Y$. Its expected value is the same as the expected value of $X,$ so we have $\operatorname E(\operatorname E(X\mid Y)) = \operatorname E(X).$

Often one sees $\displaystyle \int_S f_{X\,\mid\,Y} (x\mid y) \, dx$ but I think the notation $\displaystyle \int_S f_{X\,\mid\,Y=y} (x)\, dx$ is better.

Those a good notations; the subscripts of which you write are nonsense.