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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