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## 13.5 Categorical distribution

### 13.5.1 Probability mass functions

If $$N \in \mathbb{N}$$, $$N > 0$$, and if $$\theta \in \mathbb{R}^N$$ forms an $$N$$-simplex (i.e., has nonnegative entries summing to one), then for $$y \in \{1,\ldots,N\}$$, $\text{Categorical}(y~|~\theta) = \theta_y.$ In addition, Stan provides a log-odds scaled categorical distribution, $\text{CategoricalLogit}(y~|~\beta) = \text{Categorical}(y~|~\text{softmax}(\beta)).$ See the definition of softmax for the definition of the softmax function.

### 13.5.2 Sampling statement

y ~ categorical(theta)

Increment target log probability density with categorical_lupmf(y | theta) dropping constant additive terms.

### 13.5.3 Sampling statement

y ~ categorical_logit(beta)

Increment target log probability density with categorical_logit_lupmf(y | beta).

### 13.5.4 Stan functions

All of the categorical distributions are vectorized so that the outcome y can be a single integer (type int) or an array of integers (type int[]).

real categorical_lpmf(ints y | vector theta)
The log categorical probability mass function with outcome(s) y in $$1:N$$ given $$N$$-vector of outcome probabilities theta. The parameter theta must have non-negative entries that sum to one, but it need not be a variable declared as a simplex.

real categorical_lupmf(ints y | vector theta)
The log categorical probability mass function with outcome(s) y in $$1:N$$ given $$N$$-vector of outcome probabilities theta dropping constant additive terms. The parameter theta must have non-negative entries that sum to one, but it need not be a variable declared as a simplex.

real categorical_logit_lpmf(ints y | vector beta)
The log categorical probability mass function with outcome(s) y in $$1:N$$ given log-odds of outcomes beta.

real categorical_logit_lupmf(ints y | vector beta)
The log categorical probability mass function with outcome(s) y in $$1:N$$ given log-odds of outcomes beta dropping constant additive terms.

int categorical_rng(vector theta)
Generate a categorical variate with $$N$$-simplex distribution parameter theta; may only be used in transformed data and generated quantities blocks

int categorical_logit_rng(vector beta)
Generate a categorical variate with outcome in range $$1:N$$ from log-odds vector beta; may only be used in transformed data and generated quantities blocks