edits

Author: bmeyers

_posts/2024-11-18-Conditioning_joint_gaussian_on_sum.md

@@ -72,4 +72,4 @@ thus proving the first result.
 
 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 "near-zero" 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. 
+Assuming $\Sigma$ is rank $n$, $A$ has rank $n-1$ with exactly one "near-zero" eigenvalue. Additionally, the updated covariance become degenerate (singular), also with rank $n-1$. The updated covariance matrix is always "shrunk," so that the uncertainty is reduced in the posterior distribution.