$u$~$N(0,A)$ 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 $B=A/(A+1)$ ?

Answer 1

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.