philip_b comments on How to Throw Away Information in Causal DAGs

[philip_b]

Instead of saying ”$f\left(X\right)$ contains all information in $X$ relevant to $Y$”, it would be better to say that, $f\left(X\right)$ contains all information in $X$ that is relevant to $Y$ if you don’t condition on anything. Because it may be the case that if you condition on some additional random variable $Z$, $f\left(X\right)$ no longer contains all relevant information.

Example:

Let $X_1,X_2,Z$ be i.i.d. binary uniform random variables, i.e. each of the variables takes the value 0 with probability 0.5 and the value 1 with probability 0.5. Let $X=(X_1,X_2)$ be a random variable. Let $Y=X_1\oplus X_2\oplus Z$ be another random variable, where $\oplus$ is the xor operation. Let $f$ be the function $f\left(X\right)=f\left(\right(X_1,X_2\left)\right)=X_2$.

Then $f$ contains all information in $X$ that is relevant to $Y$. But if we know the value of $Z$, then $f$ no longer contains all information in $X$ that is relevant to $Y$.

[johnswentworth]

Good point, thanks.