Is there a difference in interpretation between $Y|X = m(X) + \epsilon$ vs. $Y = m(X) + \epsilon$?

Question

I understand that $E(Y|X)$ and $E(Y)$ are different, but difference sources, when $Y$ is a function of other random variables such as $X$, use $Y|X$ and $Y$ to describe this relationship. I'm not sure if this is a notational thing, but does something like $Y|X = m(X) + \epsilon$ and $Y = m(X) + \epsilon$ (in the context of modelling $Y$ with some function of $X$ plus random noise) have different interpretations? Assuming $X,Y,\epsilon$ are random variables.

Answer 1

There is actually no such object as $Y|X$ --- whenever this notation appears, it is an abuse of notation which operates as shorthand for specifying the conditional distribution of a random variable conditional on another random variable.$^\dagger$ Thus, the statement $Y|X = m(X) + \epsilon$ actually doesn't make any sense; the conditionality notation $|X$ is used only in the context of stipulating the distribution of a random variable, not its functional relationship to other random variables. In a regression model, you would always say $Y = m(X) + \epsilon$, not $Y|X = m(X) + \epsilon$.

When you are referring to the distribution of either $Y$ or $\epsilon$, you could refer either to the marginal distribution or the conditional given $X$. In the context of regression analysis, analysis is done conditional on the explanatory variable $X$, and so it would be usual to refer to the distribution conditional on this. The notation $Y|X$ operates as shorthand for specifying a conditional distribution. For example, the statement:

$$Y|X \sim \text{N}(m(X), \sigma_\epsilon^2),$$

is actually shorthand for the conditional distribution:

$$p(Y=y|X=x) = \text{N}(y|m(x),\sigma_\epsilon^2).$$

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$^\dagger$ Strictly speaking, it is possible to create a new object $Y|X$ which is a mapping from the range $\mathscr{X}$ to a set of random variables on different probability spaces, each conditional on the stipulated value of $X$. In most cases we do not want to bother with this, since the notation is just used as a shorthand for stipulating a conditional distribution.

Answer 2

In my opinion, the use of the equality makes dependence on $X$ explicit. The only place I have seen notation like $y \vert X$ is when the distribution of $y$ is being discussed. You see this frequently in Bayesian models like

$$ y\vert \mu , \sigma \sim \mathcal{N}(\mu, \sigma) $$

This notation tells me that there is a prior for $\sigma$ and that the distribution of $y$ depends on whatever $\sigma$ is drawn from the data generating process.

In any case, I don't see much difference between $y = $ and $y\vert X = $.