18.2 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.
18.2.1 Probability mass function
If K∈N, N∈N, and softmax(θ)∈K-simplex, then for y∈NK such that ∑Kk=1yk=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