Capitalisation and eqn formatting

src/functions-reference/bounded_discrete_distributions.Rmd

@@ -158,7 +158,7 @@ logit-scaled chance of success alpha dropping constant additive terms
 ## Binomial-logit generalized linear model (Logistic Regression) {#binomial-logit-glm}
 
 Stan also supplies a single function for a generalized linear model
-with Binomial likelihood and logit link function, i.e. a function
+with binomial likelihood and logit link function, i.e., a function
 for logistic regression with aggregated outcomes. This provides a more efficient
 implementation of logistic regression than a manually written
 regression in terms of a Binomial likelihood and matrix
@@ -167,10 +167,11 @@ multiplication.
 ### Probability mass function
 
 Suppose $N \in \mathbb{N}$, $x\in \mathbb{R}^{n\cdot m}, \alpha \in \mathbb{R}^n, \beta \in \mathbb{R}^m$, and $n \in
-\{0,\ldots,N\}$.  Then \begin{align*}
-&\text{BinomialLogitGLM}(n~|~N, x, \alpha, \beta) = \text{Binomial}(n~|~N,\text{logit}^{-1}(\alpha_i + x_i\cdot
-\beta)) \\ &= \binom{N}{n} \left( \text{logit}^{-1}(\alpha_i + \sum_{1\leq j\leq m}x_{ij}\cdot \beta_j) \right)^{n}  \left( 1 -
-\text{logit}^{-1}(\alpha_i + \sum_{1\leq j\leq m}x_{ij}\cdot \beta_j) \right)^{N - n}.  \end{align*}
+\{0,\ldots,N\}$.  Then
+\begin{align*}
+  &\text{BinomialLogitGLM}(n~|~N, x, \alpha, \beta) = \text{Binomial}(n~|~N,\text{logit}^{-1}(\alpha_i + x_i \cdot \beta)) \\
+  &= \binom{N}{n} \left( \text{logit}^{-1}(\alpha_i + \sum_{1\leq j\leq m}x_{ij}\cdot \beta_j) \right)^{n}  \left( 1 - \text{logit}^{-1}(\alpha_i + \sum_{1\leq j\leq m}x_{ij}\cdot \beta_j) \right)^{N - n}.
+\end{align*}
 
 ### Sampling statement
 
@@ -188,63 +189,63 @@ Increment target log probability density with `binomial_logit_glm_lupmf(n | N, x
 \index{{\tt \bfseries binomial\_logit\_glm\_lpmf  }!{\tt (int n \textbar\ int N, matrix x, real alpha, vector beta): real}|hyperpage}
 
 `real` **`binomial_logit_glm_lpmf`**`(int n | int N, matrix x, real alpha, vector beta)`<br>\newline
-The log Binomial probability mass of n given N trials and chance of success
+The log binomial probability mass of n given N trials and chance of success
 `inv_logit(alpha + x * beta)`.
 `r since("2.34")`
 
 <!-- real; binomial_logit_glm_lupmf; (int n | int N, matrix x, real alpha, vector beta); -->
 \index{{\tt \bfseries binomial\_logit\_glm\_lupmf  }!{\tt (int n \textbar\ int N, matrix x, real alpha, vector beta): real}|hyperpage}
 
 `real` **`binomial_logit_glm_lupmf`**`(int n | int N, matrix x, real alpha, vector beta)`<br>\newline
-The log Binomial probability mass of n given N trials and chance of success
+The log binomial probability mass of n given N trials and chance of success
 `inv_logit(alpha + x * beta)` dropping constant additive terms.
 `r since("2.34")`
 
 <!-- real; binomial_logit_glm_lpmf; (int n | int N, matrix x, vector alpha, vector beta); -->
 \index{{\tt \bfseries binomial\_logit\_glm\_lpmf  }!{\tt (int n \textbar\ int N, matrix x, vector alpha, vector beta): real}|hyperpage}
 
 `real` **`binomial_logit_glm_lpmf`**`(int n | int N, matrix x, vector alpha, vector beta)`<br>\newline
-The log Binomial probability mass of n given N trials and chance of success
+The log binomial probability mass of n given N trials and chance of success
 `inv_logit(alpha + x * beta)`.
 `r since("2.34")`
 
 <!-- real; binomial_logit_glm_lupmf; (int n | int N, matrix x, vector alpha, vector beta); -->
 \index{{\tt \bfseries binomial\_logit\_glm\_lupmf  }!{\tt (int n \textbar\ int N, matrix x, vector alpha, vector beta): real}|hyperpage}
 
 `real` **`binomial_logit_glm_lupmf`**`(int n | int N, matrix x, vector alpha, vector beta)`<br>\newline
-The log Binomial probability mass of n given N trials and chance of success
+The log binomial probability mass of n given N trials and chance of success
 `inv_logit(alpha + x * beta)` dropping constant additive terms.
 `r since("2.34")`
 
 <!-- real; binomial_logit_glm_lpmf; (array[] int n | array[] int N, row_vector x, real alpha, vector beta); -->
 \index{{\tt \bfseries binomial\_logit\_glm\_lpmf  }!{\tt (array[] int n \textbar\ array[] int N, row\_vector x, real alpha, vector beta): real}|hyperpage}
 
 `real` **`binomial_logit_glm_lpmf`**`(array[] int n | array[] int N, row_vector x, real alpha, vector beta)`<br>\newline
-The log Binomial probability mass of n given N trials and chance of success
+The log binomial probability mass of n given N trials and chance of success
 `inv_logit(alpha + x * beta)`.
 `r since("2.34")`
 
 <!-- real; binomial_logit_glm_lupmf; (array[] int n | array[] int N, row_vector x, real alpha, vector beta); -->
 \index{{\tt \bfseries binomial\_logit\_glm\_lupmf  }!{\tt (array[] int n \textbar\ array[] int N, row\_vector x, real alpha, vector beta): real}|hyperpage}
 
 `real` **`binomial_logit_glm_lupmf`**`(array[] int n | array[] int N, row_vector x, real alpha, vector beta)`<br>\newline
-The log Binomial probability mass of n given N trials and chance of success
+The log binomial probability mass of n given N trials and chance of success
 `inv_logit(alpha + x * beta)` dropping constant additive terms.
 `r since("2.34")`
 
 <!-- real; binomial_logit_glm_lpmf; (array[] int n | array[] int N, row_vector x, vector alpha, vector beta); -->
 \index{{\tt \bfseries binomial\_logit\_glm\_lpmf  }!{\tt (array[] int n \textbar\ array[] int N, row\_vector x, vector alpha, vector beta): real}|hyperpage}
 
 `real` **`binomial_logit_glm_lpmf`**`(array[] int n | array[] int N, row_vector x, vector alpha, vector beta)`<br>\newline
-The log Binomial probability mass of n given N trials and chance of success
+The log binomial probability mass of n given N trials and chance of success
 `inv_logit(alpha + x * beta)`.
 `r since("2.34")`
 
 <!-- real; binomial_logit_glm_lupmf; (array[] int n | array[] int N, row_vector x, vector alpha, vector beta); -->
 \index{{\tt \bfseries binomial\_logit\_glm\_lupmf  }!{\tt (array[] int n \textbar\ array[] int N, row\_vector x, vector alpha, vector beta): real}|hyperpage}
 
 `real` **`binomial_logit_glm_lupmf`**`(array[] int n | array[] int N, row_vector x, vector alpha, vector beta)`<br>\newline
-The log Binomial probability mass of n given N trials and chance of success
+The log binomial probability mass of n given N trials and chance of success
 `inv_logit(alpha + x * beta)` dropping constant additive terms.
 `r since("2.34")`
 
@@ -253,31 +254,31 @@ The log Binomial probability mass of n given N trials and chance of success
 \index{{\tt \bfseries binomial\_logit\_glm\_lpmf  }!{\tt (array[] int n \textbar\ array[] int N, matrix x, real alpha, vector beta): real}|hyperpage}
 
 `real` **`binomial_logit_glm_lpmf`**`(array[] int n | array[] int N, matrix x, real alpha, vector beta)`<br>\newline
-The log Binomial probability mass of n given N trials and chance of success
+The log binomial probability mass of n given N trials and chance of success
 `inv_logit(alpha + x * beta)`.
 `r since("2.34")`
 
 <!-- real; binomial_logit_glm_lupmf; (array[] int n | array[] int N, matrix x, real alpha, vector beta); -->
 \index{{\tt \bfseries binomial\_logit\_glm\_lupmf  }!{\tt (array[] int n \textbar\ array[] int N, matrix x, real alpha, vector beta): real}|hyperpage}
 
 `real` **`binomial_logit_glm_lupmf`**`(array[] int n | array[] int N, matrix x, real alpha, vector beta)`<br>\newline
-The log Binomial probability mass of n given N trials and chance of success
+The log binomial probability mass of n given N trials and chance of success
 `inv_logit(alpha + x * beta)` dropping constant additive terms.
 `r since("2.34")`
 
 <!-- real; binomial_logit_glm_lpmf; (array[] int n | array[] int N, matrix x, vector alpha, vector beta); -->
 \index{{\tt \bfseries binomial\_logit\_glm\_lpmf  }!{\tt (array[] int n \textbar\ array[] int N, matrix x, vector alpha, vector beta): real}|hyperpage}
 
 `real` **`binomial_logit_glm_lpmf`**`(array[] int n | array[] int N, matrix x, vector alpha, vector beta)`<br>\newline
-The log Binomial probability mass of n given N trials and chance of success
+The log binomial probability mass of n given N trials and chance of success
 `inv_logit(alpha + x * beta)`.
 `r since("2.34")`
 
 <!-- real; binomial_logit_glm_lupmf; (array[] int n | array[] int N, matrix x, vector alpha, vector beta); -->
 \index{{\tt \bfseries binomial\_logit\_glm\_lupmf  }!{\tt (array[] int n \textbar\ array[] int N, matrix x, vector alpha, vector beta): real}|hyperpage}
 
 `real` **`binomial_logit_glm_lupmf`**`(array[] int n | array[] int N, matrix x, vector alpha, vector beta)`<br>\newline
-The log Binomial probability mass of n given N trials and chance of success
+The log binomial probability mass of n given N trials and chance of success
 `inv_logit(alpha + x * beta)` dropping constant additive terms.
 `r since("2.34")`