Categorical Distribution - Properties

Properties

• The distribution is completely given by the probabilities associated with each number i:, i = 1,...,k, where . The possible probabilities are exactly the standard -dimensional simplex; for k = 2 this reduces to the possible probabilities of the Bernoulli distribution being the 1-simplex,

• The distribution is a special case of a "multivariate Bernoulli distribution" in which exactly one of the k 0-1 variables takes the value one.

• Let be the realisation from a categorical distribution. Define the random vector Y as composed of the elements:

where I is the indicator function. Then Y has a distribution which is a special case of the multinomial distribution with parameter . The sum of independent and identically distributed such random variables Y constructed from a categorical distribution with parameter is multinomially distributed with parameters and

• The conjugate prior distribution of a categorical distribution is a Dirichlet distribution. See the section below for more discussion.

• 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 indicator function of an observation having a value i, equivalent to the Iverson bracket function or the Kronecker delta function is Bernoulli distributed with parameter