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

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.

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:

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

$$^\dagger$$ 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.

• An example of $Y|X$ appears at the bottom of page 12, I assume I can ignore the "$|X$" in this case? In general, $Y|X$ should only occur in the context of referencing a conditional probability density or mass function for $Y$ given $X=x$? Mar 10, 2020 at 3:00
• Yes, I would just ignore it.
– Ben
Mar 10, 2020 at 3:17

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 $$y\vert X =$$.