15.1 Multinomial distribution

15.1.1 Probability mass function

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

15.1.2 Sampling statement

y ~ multinomial(theta)

Increment target log probability density with multinomial_lupmf(y | theta).

15.1.3 Stan functions

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

real multinomial_lupmf(int[] y | vector theta)
The log multinomial probability mass function with outcome array y of size \(K\) given the \(K\)-simplex distribution parameter theta and (implicit) total count N = sum(y) dropping constant additive terms

int[] multinomial_rng(vector theta, int N)
Generate a multinomial variate with simplex distribution parameter theta and total count \(N\); may only be used in transformed data and generated quantities blocks