18.2 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.
18.2.1 Probability mass function
If \(K \in \mathbb{N}\), \(N \in \mathbb{N}\), and \(\text{softmax}(\theta) \in \text{$K$-simplex}\), then for \(y \in \mathbb{N}^K\) such that \(\sum_{k=1}^K y_k = N\), \[ \text{MultinomialLogit}(y \mid \gamma) = \text{Multinomial}(y \mid \text{softmax}(\gamma)) = \binom{N}{y_1,\ldots,y_K} \prod_{k=1}^K [\text{softmax}(\gamma_k)]^{y_k}, \] where the multinomial coefficient is defined by \[ \binom{N}{y_1,\ldots,y_k} = \frac{N!}{\prod_{k=1}^K y_k!}. \]
18.2.2 Sampling statement
y ~ multinomial_logit(gamma)
Increment target log probability density with multinomial_logit_lupmf(y | gamma).
Available since 2.24
18.2.3 Stan functions
real multinomial_logit_lpmf(array[] int y | vector gamma)
The log multinomial probability mass function with outcome array y
of size \(K\) given the log \(K\)-simplex distribution parameter \(\gamma\) and
(implicit) total count N = sum(y)
Available since 2.24
real multinomial_logit_lupmf(array[] int y | vector gamma)
The log multinomial probability mass function with outcome array y
of size \(K\) given the log \(K\)-simplex distribution parameter \(\gamma\) and (implicit) total count N = sum(y) dropping constant additive
terms
Available since 2.25
array[] int multinomial_logit_rng(vector gamma, int N)
Generate a variate from a multinomial distribution with probabilities
softmax(gamma) and total count N; may only be used in transformed data and
generated quantities blocks.
Available since 2.24