cleanup initial row/col stochastic matrix docs

Author: SteveBronder

src/reference-manual/transforms.qmd

@@ -672,14 +672,14 @@ z_k
 .
 $$
 
-## Column Stochastic Matrix {#column-stochastic-matrix-transform.section}
+## Stochastic Matrix {#stochastic-matrix-transform.section}
 
-The `column_stochastic_matrix[N, M]` type in Stan represents an \(N \times M\) 
-matrix where each column is a unit simplex of dimension \(N\). In other words, 
-each column of the matrix is a vector constrained to have non-negative entries 
-that sum to one.
+The `column_stochastic_matrix[N, M]` and `row_stochastic_matrix[N, M]` type in 
+Stan represents an \(N \times M\) matrix where each column(row) is a unit simplex 
+of dimension \(N\). In other words, each column(row) of the matrix is a vector 
+constrained to have non-negative entries that sum to one.
 
-### Definition of a Column Stochastic Matrix {-}
+### Definition of a Stochastic Matrix {-}
 
 A column stochastic matrix \(X \in \mathbb{R}^{N \times M}\) is defined such 
 that for each column \(j\) (where \(1 \leq j \leq M\)):
@@ -694,43 +694,28 @@ $$
 \sum_{i=1}^N X_{ij} = 1.
 $$
 
-This definition ensures that each column of the matrix \(X\) lies on the 
-\(N-1\) dimensional unit simplex, similar to the `simplex[N]` type, but 
-extended across multiple columns.
-
-### Inverse Transform for Column Stochastic Matrix {-}
-
-The inverse transform for the `column_stochastic_matrix` type is an extension 
-of the unit simplex inverse transform to multiple columns. The process can be 
-understood by applying the stick-breaking metaphor independently to each column.
+A row stochastic matrix is defined similarly but with the axis flipped such
+that 
 
-For each column \(j\) of the matrix \(X\), an unconstrained vector \(y_j \in 
-\mathbb{R}^{N-1}\) is mapped to the column \(X_{\cdot j}\) on the unit simplex 
-using the following steps:
-
-1.  Begin with a stick of unit length.
-2.  Break off a piece corresponding to each element \(X_{ij}\) of the column 
-vector, where the size of each piece is determined by an intermediate value 
-\(z_{ij}\), which is itself derived from the unconstrained 
-parameter \(y_{ij}\).
-3.  The intermediate vector \(z_j \in \mathbb{R}^{N-1}\) is 
-defined elementwise for each \(j\) and for \(1 \leq i < N\) by
 
 $$
-z_{ij} = \mathrm{logit}^{-1}\left( y_{ij} + \log\left(\frac{1}{N - i}\right) \right).
+X_{ij} \geq 0 \quad \text{for } 1 \leq j \leq N,
 $$
 
-4.  The stick sizes \(X_{ij}\) for \(1 \leq i < N\) are then calculated recursively by
+and
 
 $$
-X_{ij} = \left( 1 - \sum_{i'=1}^{i-1} X_{i'j} \right) z_{ij}.
+\sum_{j=1}^N X_{ij} = 1.
 $$
 
-5.  The last element of each column \(X_{Nj}\) is set to the length of the remaining piece of the stick:
+This definition ensures that each column(row) of the matrix \(X\) lies on the 
+\(N-1\) dimensional unit simplex, similar to the `simplex[N]` type, but 
+extended across multiple columns(rows).
 
-$$
-X_{Nj} = 1 - \sum_{i=1}^{N-1} X_{ij}.
-$$
+### Inverse Transform for Stochastic Matrix {-}
+
+For the column and row stochastic matrices the inverse transform is the same
+as simplex, but applied to each column(row).
 
 ### Absolute Jacobian Determinant for the Inverse Transform {-}
 
@@ -742,37 +727,17 @@ $$
 \left| \det J_j \right| = \prod_{i=1}^{N-1} \left( z_{ij} (1 - z_{ij}) \left( 1 - \sum_{i'=1}^{i-1} X_{i'j} \right) \right).
 $$
 
-Thus, the overall Jacobian determinant for the entire `column_stochastic_matrix` 
-is the product of the determinants for each column:
+Thus, the overall Jacobian determinant for the entire `column_stochastic_matrix` and `row_stochastic_matrix` 
+is the product of the determinants for each column(row):
 
 $$
 \left| \det J \right| = \prod_{j=1}^{M} \left| \det J_j \right|.
 $$
 
-### Transform for Column Stochastic Matrix {-}
-
-To transform from a column stochastic matrix \(X\) back to the unconstrained space \(y_j\) 
-for each column \(j\):
-
-1. The break proportions \(z_{ij}\) are first determined by
-
-$$
-z_{ij} = \frac{X_{ij}}{1 - \sum_{i'=1}^{i-1} X_{i'j}}.
-$$
-
-2. The corresponding unconstrained parameters \(y_{ij}\) are then computed as
-
-$$
-y_{ij} = \mathrm{logit}(z_{ij}) - \log\left(\frac{1}{N - i}\right).
-$$
-
-By applying this process to each column \(j\) independently, the entire column 
-stochastic matrix \(X\) can be transformed to or from the unconstrained space.
-
-This formulation allows the `column_stochastic_matrix[N, M]` type to be used 
-effectively in models requiring columns of a matrix to be unit simplexes, 
-such as in multi-category probability models or compositional data analysis.
+### Transform for Stochastic Matrix {-}
 
+For the column and row stochastic matrices the transform is the same
+as simplex, but applied to each column(row).
 
 ## Unit vector {#unit-vector.section}
 

src/reference-manual/types.qmd

@@ -726,12 +726,8 @@ row is a 4-simplex, is declared as:
 row_stochastic_matrix[3, 4] theta;
 ```
 
-The `column_stochastic_matrix` type is implemented as a matrix where each 
-column is individually constrained to be a simplex. This means that each 
-column must be a valid simplex with non-negative elements that sum to 1.
-
-As with simplexes, `column_stochastic_matrix` variables are subject to 
-validation, ensuring that each column satisfies the simplex constraints. 
+As with simplexes, `column(row)_stochastic_matrix` variables are subject to 
+validation, ensuring that each column(row) satisfies the simplex constraints. 
 This validation accounts for floating-point imprecision, with checks 
 performed up to a statically specified accuracy threshold \(\epsilon\).