additional discussion

Author: bmeyers

_posts/2024-11-18-Conditioning_joint_gaussian_on_sum.md

@@ -69,4 +69,6 @@ thus proving the first result.
 
 ### Discussion
 
-Note that both $v$ and $A$ do not depend at all on $s$ and can be pre-calculated. Also, because $\Sigma$ is positive-semidefinite, the entries of $v$ are all on the interval $[0,1]$. $A$ has rank $n-1$ with exactly one negative eigenvalue, and so the covariance is always shrunk, but in a non-obvious ways that depends on the original covariance matrix.
+Note that both $v$ and $A$ do not depend at all on $s$ and can be pre-calculated. The entries of $v$ are all on the interval $[0,1]$ and, in fact, form a simplex (their values sum to $1$). The posterior mean is always updated to be exactly consistent with the observed sum.
+
+$A$ has rank $n-1$ with exactly one negative eigenvalue, and so the covariance is always shrunk in a way that depends on the original covariance matrix. This has the effect in which blocks of more highly correlated variable will have their variance (diagonal entries) shrunk more than independent variables.