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15.2 Multinomial Distribution, Logit Parameterization

Stan also provides a version of the multinomial probability mass function distribution with the $\text{$K$-simplex}$ for the event count probabilities per category given on the unconstrained logistic scale.

15.2.1 Probability Mass Function

If $K \in \mathbb{N}$, $N \in \mathbb{N}$, and $\text{softmax}^{-1}(\theta) \in \text{$K$-simplex}$, then for $y \in \mathbb{N}^K$ such that $\sum_{k=1}^K y_k = N$, $$\text{MultinomialLogit}(y|\theta) = \text{Multinomial}(y|\text{softmax}^{-1}(\theta)) = \binom{N}{y_1,\ldots,y_K} \prod_{k=1}^K [\text{softmax}^{-1}(\theta)_k]^{y_k},$$ where the multinomial coefficient is defined by $$\binom{N}{y_1,\ldots,y_k} = \frac{N!}{\prod_{k=1}^K y_k!}.$$

15.2.2 Sampling Statement

y ~ multinomial_logit(theta)

Increment target log probability density with multinomial_logit_lpmf(y | theta) dropping constant additive terms.

15.2.3 Stan Functions

real multinomial_logit_lpmf(int[] y | vector theta)
The log multinomial probability mass function with outcome array y of size $K$ given the $K$-simplex distribution parameter $\text{softmax}^{-1}(\theta)$ and (implicit) total count N = sum(y)

int[] multinomial_logit_rng(vector theta, int N)
Generate a multinomial variate with simplex distribution parameter $\text{softmax}^{-1}(\theta)$ and total count $N$; may only be used in transformed data and generated quantities blocks