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Why does the formula $\operatorname{Var} (Y\mid X)= E \left (\left(Y- E (Y\mid X)\right )^2\mid X\right )$ condition $EY$, but not $Y$, on $X$?

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    $\begingroup$ I don't understand the question, because by definition "$EY$" is a number. What, then, do you mean by "condition $EY$" on anything? $\endgroup$
    – whuber
    Commented Oct 12, 2023 at 11:44

1 Answer 1

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It does condition $Y$ on $X$. But only after centering $Y$ around $E\left( Y \mid X \right)$.

$$\begin{align*} \tilde Y &= \left( Y - E\left( Y \mid X = x \right) \right) ^2 \\ V(Y \mid X = x) &= E\left( \tilde Y \mid X = x \right) \end{align*}$$

First we compute squared deviations of $Y$ itself around the conditional expectation. Then we compute the conditional expectation of those squared deviations.

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