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Before we discuss the Dirichlet distribution, we define the Beta distribution.

Definition 2.22 (Beta distribution) A random variable is said to have a Beta distribution with parameters and if it has a pdf defined as:

where and .

The Dirichlet distribution , named after Johann Peter Gustav Lejeune Dirichlet (1805–1859), is a multivariate distribution parameterized by a vector of positive parameters .

Specifically, the joint density of an ‐dimensional random vector is defined as:

where is an indicator function.

Definition 2.23 (Indicator function) The indicator function of a subset of a set is a function

defined as

The components of the random vector thus are always positive and have the property . The normalizing constant is the multinomial beta function, that is defined as:

where we used the notation and for the Gamma function.

Because the Dirichlet distribution creates positive numbers that always sum to 1, it is extremely useful to create candidates for probabilities of possible outcomes. This distribution is very popular and related to the multinomial distribution which needs numbers summing to 1 to model the probabilities in the distribution. The multinomial distribution is defined in Section 2.3.2.

With the notation mentioned above and as the sum of all parameters, we can calculate the moments of the distribution. The first moment vector has coordinates:

The covariance matrix has elements:

and when

The covariance matrix is singular (its determinant is zero).

Finally, the univariate marginal distributions are all beta with parameters . All these are in the reference (see Balakrishnan and Nevzorov 2004).

Please refer to Lin (2016) for the proof of the properties of the Dirichlet distribution.