Bayesian update vs optimization Say I have a multivariate normal vector 
$$ r \sim N(\mu , \Sigma ) \Rightarrow Pr \sim N(P\mu , P'\Sigma P ) $$ 
and I observe that 
$$  Pr = Q $$ 
Now I can use Bayes rule to calculate the updated mean of $r$ using the formula mentioned here
$$ E(r | Pr = Q) = \mu + \Sigma P'[ P \Sigma^{-1}P']^{-1
}[Q - P\mu] \quad \quad (i) $$
This is the same expression as would be derived if you minimized 
\begin{align}
 \min_{\tilde r} & ( \tilde r - \mu)'\Sigma^{-1}( \tilde r-\mu) \quad \quad (ii)\\
\text{subject to } & P \tilde r = Q 
\end{align}
My question: why does the constrained minimization (ii) give the same result as the bayesian updating case (i)?  Is there a conceptual link? I think it is related to MLE or MAP estimation but havent been able to link the two together.
 A: Bayes rule is essentially function composition of probability distribution functions. It doesn't inherently give you any parameters, it gives you a new (AKA posterior) distribution.
Now, you're trying to find a posterior that is normally distributed. In this case, it's possible, but it's not a given from Bayes alone; it just so happens that Normal Marginal + Normal Prior results in a distribution that is also Normal.
A Normal distribution is defined by having a single maximum probability mass at the mean, and the rest of the probability mass decaying exponentially symmetrically from that. 
So, to find the mean of any distribution you already know is Normal, you just need to find the maximum of the posterior probability density function... 'of/from the posterior' being the English equivalent of 'a posteriori'. So there's your first conceptual link, to MAP. MLE is just MAP without a strong opinion about the prior distribution.
But that doesn't answer the question why they're equivalent. Another way of looking at the problem is to flip the perspective. What is the most likely is what is the least unlikely, since the total probability has to sum to 1.
This means you can find the most likely location of the parameters by minimizing error, where the error is some metric of the difference between your current guess and the data - same general idea as least-squares, essentially.
An estimator that does that, i.e. minimizes expected error of the posterior, is the definition of a Bayes estimator, which gives you (i) for multivariate Normal. It is also exactly what you're doing in (ii).
