suppose we have two independent r.v. $Z$(standard-normal) and $X$(any r.v.) with $E[X^2]=\sigma^2$. Let \begin{align*} Y=X+Z \end{align*}
My question is MMSE given by \begin{align*} \mathbb E[(X-\mathbb E[X|Y])^2] \end{align*}
is it a decreasing( or non-increasing) function of $\sigma^2$?
My intuition tells me is that if we increase $\sigma^2$ then MMSE should decrease. But how to show it formally?
Thanks for any advice.