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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