Categorical Distribution

Categorical Distribution is one of the generalization of Bernoulli distribution. It is also called generalized Bernoulli distribution, multinoulli distribution.

• $K \in \mathbb{N}$,

\[\Delta_{K} := \{ (p_{1}, \ldots, p_{K}) \in [0,1]^{K} \mid \sum_{i=1}^{K} p_{i} = 1 \} .\]

$\mathrm{Categ}(x; p)$ is a p.d.f. of categorical distribution given $p \in \Delta_{K}$ defined by

\[x \in \{1, \ldots, K\}, \ \mathrm{Categ}(x; p) := \sum_{i=1}^{K} 1_{i = x} p_{i} .\]

Reference

• Categorical distribution - Wikipedia