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 Aug 20 comment Signal extraction from multivariate normal to be clear, you have implicitly done the following: $E[\theta|y]=E[\theta] + E[\theta-\bar{\theta}|y-\bar{y}] = E[\theta] + E[\theta-\bar{\theta}|y-\bar{\theta}-a]$ and I am unclear what axioms of expected probability where involved in the process. Then, in eq. $a=a^e$ Aug 20 awarded Supporter Aug 20 comment Signal extraction from multivariate normal I meant substitute $y=\theta+\varepsilon+a$ for $y$ and you are done. Aug 20 comment Signal extraction from multivariate normal Many thanks, this is very useful. I was thinking precisely along those line i.e. centering the variables around zero. What I am not so clear is on the jump from $E[\theta^e|y]$ to $E[\theta^e|y]=\bar{\theta} + E[c|x]$. What is the general rule that allows you to do this? How was the $\bar{\theta}$ factored out? I understand the intuition, but not the formal derivation. BTW after the third equal sign in the last derivation just substitute $y=\theta^e+\varepsilon+a^e$ for $y$ and you are done. Aug 20 awarded Teacher Aug 20 answered How do I express this chain rule? Aug 20 awarded Editor Aug 20 revised Signal extraction from multivariate normal added 6 characters in body Aug 20 awarded Student Aug 20 comment Signal extraction from multivariate normal @Michael Hardy All expected values have a bar, as in $\bar{\theta}$, except for the definition of the mean in $\mathcal{N}(\theta,\sigma_{\theta}^2)$ where I corrected typo. Aug 20 revised Signal extraction from multivariate normal edited tags Aug 20 asked Signal extraction from multivariate normal