Generalized Dirichlet Distribution

In statistics, the generalized Dirichlet distribution (GD) is a generalization of the Dirichlet distribution with a more general covariance structure and almost twice the number of parameters. Random variables with a GD distribution are neutral.

The density function of is

$\left[ \prod_{i=1}^{k-1}B(a_i,b_i)\right]^{-1} p_k^{b_{k-1}-1} \prod_{i=1}^{k-1}\left[ p_i^{a_i-1}\left(\sum_{j=i}^kp_j\right)^{b_{i-1}-(a_i+b_i)}\right]$

where we define . Here denotes the Beta function. This reduces to the standard Dirichlet distribution if for ( is arbitrary).

Wong gives the slightly more concise form for

$\prod_{i=1}^k\frac{x_i^{\alpha_i-1}\left(1-x_1-\ldots-x_i\right)^{\gamma_i}}{B(\alpha_i,\beta_i)}$

where for and . Note that Wong defines a distribution over a dimensional space (implicitly defining ) while Connor and Mosiman use a dimensional space with . The remainder of this article will use Wong's notation.

Read more about Generalized Dirichlet Distribution: General Moment Function, Reduction To Standard Dirichlet Distribution, Bayesian Analysis, Sampling Experiment

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