$u$~$N(0,A)$ and z$|u$~$N(u,1)$ how to show that $u|z$~$N(Bz,B)$ where $B=A/(A+1)$?

Question

$u$~$N(0,A)$ and $z|u$~$N(u,1)$ how to show that $u|z$~$N(Bz,B)$ where $B=A/(A+1)$ ?

Empirical Bayes and Large Scale Interference

Knowing $f_U$ and $f_{Z\mid U}$ yields $f_{U,Z}$, which yields $f_{U\mid Z}$. Which part of this program have you trouble with?
Never wrote $f(U,Z)$, but $f_{U,Z}$ the joint density of $(U,Z)$.

Patrick Li · Answer 1 · 2013-01-28 16:12:54Z

up vote 1 down vote

Note that

$$p(\mu|z) \propto p(z|\mu)p(\mu)$$

After you plug it $p(z|\mu)$ and $p(\mu)$ and you will find $p(\mu|z)$ has a pdf that is normal. And you can easily derive its mean and variance.

answered Jan 28 at 16:12

Patrick Li
2,3321624

	but what is $p(z)$ ? $N(0,1)$ ? – Qbik Jan 28 at 16:22
	@Qbik There is no $p(z)$ involved here. You only need the conditional distribution of $p(z\|\mu)$. – Patrick Li Jan 28 at 16:55

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