# Deviation due to conditioning

Let $A$ and $B$ be random variables. Can we upper-bound the following expression? $$\mathbb{E}\Big[\Big(\mathbb{E}[A|B] - \mathbb{E}[A]\Big)^2\Big]$$ The above looks classical research. However, I am not able to find any references. Is there any literature which deals with such expressions?

• This is bounded by $var(A)$ since this is the variance of the conditional expectation. – Xi'an Apr 30 '16 at 13:41

$E[E[A|B]]=E[A]$ and so the expression is simply the variance of $E[A|B]$ \begin{align*} \mathrm{Var}(A) &= E[\mathrm{Var}(A|B)] + \mathrm{Var}(E[A|B])\\ \mathrm{Var}(E[A|B]) &= \mathrm{Var}(A) -E[\mathrm{Var}(A|B)]\\ \implies \mathrm{Var}(E[A|B]) &\leq \mathrm{Var}(A) \end{align*}