Quote by kasraa
Part one:
The posterior [tex] p \left( xZ \right) [/tex], has a mean and a (co)variance. Its mean is the MMSE estimator, [tex] E \left[ xZ \right] [/tex], and its variance (or the trace of its covariance matrix, if it's a random vector) is the minimum mean squared error. Am I right?
Thanks.

I don't think so. For a random vector of observations, the MMSE for the posterior estimate is the
minimized trace of the covariance matrix. This is consistent with the discussion in the link I provided. As for the rest, I'm not following you. I don't understand why you're double conditioning on Z for instance. Someone else will have to try and help you