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In statistics, the generalized Dirichlet distribution is a generalization of the Dirichlet distribution with a more general covariance structure and almost twice the number of parameters. Random vectors with a GD distribution are completely neutral.
The density function of p 1 , … , p k − 1 {\displaystyle p_{1},\ldots ,p_{k-1}} is
where we define p k = 1 − ∑ i = 1 k − 1 p i {\textstyle p_{k}=1-\sum _{i=1}^{k-1}p_{i}}. Here B {\displaystyle B} denotes the Beta function. This reduces to the standard Dirichlet distribution if b i − 1 = a i + b i {\displaystyle b_{i-1}=a_{i}+b_{i}} for 2 ⩽ i ⩽ k − 1 {\displaystyle 2\leqslant i\leqslant k-1} .
For example, if k=4, then the density function of p 1 , p 2 , p 3 {\displaystyle p_{1},p_{2},p_{3}} is