Say I have a multivariate normal vector
$r$~$N(\mu , \Sigma ) => Pr$~$N(P\mu , P'\Sigma P )$
and I observe that
$ Pr = Q$
Now I can use Bayes rule to calculate the updated mean of $r$ using the formula mentioned here
$E(r/ (Pr =Q)) = \mu + \Sigma P'[ P \Sigma^{-1}P']^{-1 }[Q - P\mu]$......................................(i)
This is the same expression as would be derived if you minimized
$(r - \mu)'\Sigma^{-1}(r-\mu)$ subject to $Pr = Q $.............(ii)
My question is- why does the constrained minimization (ii) give the same result as the bayesian updating case (i)? Is there a conceptual link?