Categorical distribution

A categorical distribution is the most general distribution whose sample space is the set {1, 2, ..., "n"}.

It is the generalization of the Bernoulli distribution for a
categorical random variable.

It should not be confused with the multinomial distribution.

Consider the extended analogy:
:Bernoulli random variable : Bernoulli distribution : Binomial distribution :Categorical random variable : Categorical distribution : Multinomial distribution

The distribution is completely given by the probabilities associated with each number "k": $p\_k\; =\; P(X\; =\; k)$, "k" = 1,...,"n".

In the BUGS software, this distribution is represented as:

BUGS: r ~ dcat(p [] )// BUGS assigns categories the values 1, 2, ..., K

The sufficient statistic from "n" independent observations is the set of counts (or,equivalently, proportion) of observations in each category, where the total number of trials (="n") is fixed.

The conjugate prior is the Dirichlet distribution.