Multivariate Discrete Distributions

The multivariate discrete distributions are over multiple integer values, which are expressed in Stan as arrays.

Multinomial distribution

Probability mass function

If $K\in \mathbb{N}$, $N\in \mathbb{N}$, and $\theta \in K\text{-simplex}$, then for $y\in {\mathbb{N}}^K$ such that $\sum _{k=1}^K{y}_k=N$, $$\text{Multinomial}(y|\theta )=(\genfrac{}{}{0ex}{}{N}{y_1,\dots ,y_K})\prod _{k=1}^K{\theta }_k^{y_k},$$ where the multinomial coefficient is defined by $$(\genfrac{}{}{0ex}{}{N}{y_1,\dots ,y_k})=\frac{N!}{\prod _{k=1}^K{y}_k!}.$$

Distribution statement

y ~ multinomial(theta)

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

Available since 2.0

Stan functions

real multinomial_lpmf(array[] 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)

Available since 2.12

real multinomial_lupmf(array[] 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

Available since 2.25

array[] 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

Available since 2.8

Multinomial distribution, logit parameterization

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

Probability mass function

If $K\in \mathbb{N}$, $N\in \mathbb{N}$, and $\text{softmax}(\theta )\in K\text{-simplex}$, then for $y\in {\mathbb{N}}^K$ such that $\sum _{k=1}^K{y}_k=N$, $$\begin{array}{rl}\text{MultinomialLogit}(y\mid \gamma )& =\text{Multinomial}(y\mid \text{softmax}(\gamma ))\\ & =(\genfrac{}{}{0ex}{}{N}{y_1,\dots ,y_K})\prod _{k=1}^K[\text{softmax}({\gamma }_k){]}^{y_k},\end{array}$$ where the multinomial coefficient is defined by $$(\genfrac{}{}{0ex}{}{N}{y_1,\dots ,y_k})=\frac{N!}{\prod _{k=1}^K{y}_k!}.$$

Distribution statement

y ~ multinomial_logit(gamma)

Increment target log probability density with multinomial_logit_lupmf(y | gamma).

Available since 2.24

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

Dirichlet-multinomial distribution

Stan also provides the Dirichlet-multinomial distribution, which generalizes the Beta-binomial distribution to more than two categories. As such, it is an overdispersed version of the multinomial distribution.

Probability mass function

If $K\in \mathbb{N}$, $N\in \mathbb{N}$, and $\alpha \in {\mathbb{R}}_{+}^K$, then for $y\in {\mathbb{N}}^K$ such that $\sum _{k=1}^K{y}_k=N$, the PMF of the Dirichlet-multinomial distribution is defined as $$\text{DirMult}(y|\theta )=\frac{\mathrm{\Gamma }({\alpha }_0)\mathrm{\Gamma }(N+1)}{\mathrm{\Gamma }(N+{\alpha }_0)}\prod _{k=1}^K\frac{\mathrm{\Gamma }(y_k+{\alpha }_k)}{\mathrm{\Gamma }({\alpha }_k)\mathrm{\Gamma }(y_k+1)},$$ where ${\alpha }_0$ is defined as ${\alpha }_0=\sum _{k=1}^K{\alpha }_k$.

Distribution statement

y ~ dirichlet_multinomial(alpha)

Increment target log probability density with dirichlet_multinomial_lupmf(y | alpha).

Available since 2.34

Stan functions

real dirichlet_multinomial_lpmf(array[] int y | vector alpha)
The log multinomial probability mass function with outcome array y with $K$ elements given the positive $K$-vector distribution parameter alpha and (implicit) total count N = sum(y).

Available since 2.34

real dirichlet_multinomial_lupmf(array[] int y | vector alpha)
The log multinomial probability mass function with outcome array y with $K$ elements, given the positive $K$-vector distribution parameter alpha and (implicit) total count N = sum(y) dropping constant additive terms.

Available since 2.34

array[] int dirichlet_multinomial_rng(vector alpha, int N)
Generate a multinomial variate with positive vector distribution parameter alpha and total count N; may only be used in transformed data and generated quantities blocks. This is equivalent to multinomial_rng(dirichlet_rng(alpha), N).

Available since 2.34