16.2 Multinomial distribution, logit parameterization
Stan also provides a version of the multinomial probability mass function distribution with the K-simplex for the event count probabilities per category given on the unconstrained logistic scale.
16.2.1 Probability mass function
If K∈N, N∈N, and softmax−1(θ)∈K-simplex, then for y∈NK such that ∑Kk=1yk=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!}.
16.2.2 Sampling statement
y ~ multinomial_logit(theta)
Increment target log probability density with multinomial_logit_lupmf(y | theta).
Available since 2.24
16.2.3 Stan functions
real multinomial_logit_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 \text{softmax}^{-1}(\theta) and
(implicit) total count N = sum(y)
Available since 2.24
real multinomial_logit_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 \text{softmax}^{-1}(\theta) and (implicit) total count N = sum(y) dropping constant additive
terms
Available since 2.25
array[] 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
Available since 2.24