Merge pull request #841 from stan-dev/update-sum-to-zero

fix talk about correlation and typo

Author: bob-carpenter

src/reference-manual/transforms.qmd

@@ -477,14 +477,13 @@ $$
 For the transform, Stan uses the first part of an isometric log ratio
 transform; see [@egozcue+etal:2003] for the basic definitions and
 Chapter 3 of [@filzmoser+etal:2018] for the pivot coordinate version
-used here.  Stan uses the isometric log ratio transform because it
-induces a geometry with zero correlation among the dimensions, making
-it easier for HMC to explore than simpler alternatives such as setting
-the final element to the negative sum of the first elements; see, e.g.,
-[@seyboldt:2024].
-
-
-
+used here.  Stan uses the isometric log ratio transform because it 
+results in equal variances of the the constrained sum to zero
+vector see, e.g.,[@seyboldt:2024]. Simpler alternatives, such as setting the
+final element to the negative sum of the first elements, do not result in
+equal variances. The $N - 1$ unconstrained parameters are independent, however, 
+the sum-to-zero constraint induces a negative correlation across the 
+constrained vector values.
 
 ### Zero sum transform {-}
 
@@ -499,7 +498,7 @@ S_N = 0
 $$
 and
 $$
-y_N = -x_{N + 1} \cdot \frac{\sqrt{N \cdot (N + 1)}{N}}.
+y_N = -x_{N + 1} \cdot \sqrt{1 + \frac{1}{N}}.
 $$
 The for each $n$ from $N - 1$ down to $1$, let
 $$