How to Calculate the Dirichlets Multinomial Distribution - Definition, Formula and Example

Definition of Dirichlets Multinomial Distribution:

Dirichlet-multinomial distribution is the probability of a data set which contain some individual vector variables whose value is undeterminant. It is also called the Dirichlet compound multinomial distribution (DCM) or multivariate Polya distribution

Formula :

Where,

Pr(Z | α ) is Dirichlets Multinominal Distribution A is ∑_k α_k (Sigma) represents summation of Parameter vector (α) values N is ∑_k n_k represents summation of Independent trial (n) values k is number of trials Γ (gamma) represents factorial function of (x-1) Π (pi) represents product function

Example :

Calculate Dirichlets Multinominal Distribution where the Number of Trials = 2 and Independent trial values and parameter vector values are listed below: Independent Trials (n₁) = 2 Parameter Vector (α₁) = 5 Independent Trials (n₂) = 3 Parameter Vector (α₂) = 8

Solution:

Calculate A and N values A = ∑_k α_k = 5 + 8 = 13 N = ∑_k n_k = 2 + 3 = 5 Substitute values in Formula: Dirichlets Multinomial = = [Γ(13) / Γ( 5+13)] X [Γ(2+5) / Γ(5)] X [Γ(3+8) / Γ(8)] = [(13 - 1)! / (18 - 1)!] X [(7 - 1)! / (5 - 1)!] X [(11 - 1)! / (8 - 1)!] = (12! / 17!) X (6! / 4!) X (10! / 7!) = 21600 / 742560 Dirichlets Multinomial = 0.0291