Generalized Dirichlet distribution
From Wikipedia, the free encyclopedia
In statistics, the generalized Dirichlet distribution (GD) is a generalization of the Dirichlet distribution with a more general covariance structure and twice the number of parameters. Random variables with a GD distribution are neutral[1].
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]](../../../../math/8/6/7/8670037fd9a237a6fd2d574253e5e345.png)
where we define 
. Here B(x,y) denotes the Beta function. This reduces to the standard Dirichlet distribution if bi − 1 = ai + bi for 
(b0 is arbitrary).
Wong [2] gives the slightly more concise form for 
where γi = βj − αj + 1 − βj + 1 for 
and γk = βk − 1. Note that Wong defines a distribution over a k dimensional space (implicitly defining 
) while Connor and Mosiman use a k − 1 dimensional space with 
. The remainder of this article will use Wong's notation.
Contents |
[edit] General moment function
If 
, then
![E\left[X_1^{r_1}X_2^{r_2}\cdots X_k^{r_k}\right]=
\prod_{j=1}^k
\frac{
\Gamma\left(\alpha_j+\beta_j\right)
\Gamma\left(\alpha_j+r_j\right)
\Gamma\left(\beta_j+\delta_j\right)
}{
\Gamma\left(\alpha_j\right)
\Gamma\left(\beta_j\right)
\Gamma\left(\alpha_j+\beta_j+r_j+\delta_j\right)
}](../../../../math/a/2/6/a26d22d80bc59c05cf6e5da2e4bbbb5e.png)
where 
. Thus
[edit] Reduction to standard Dirichlet distribution
As stated above, if bi − 1 = ai + bi for 
then the distribution reduces to a standard Dirichlet. This condition is different from the usual case in which the new parameters being equal to zero gives the original distribution. However, in the case of the GDD attempting to do this results in a very complicated density function.
[edit] Bayesian analysis
Suppose is generalized Dirichlet, and that Y | X is multinomial with n trials (here 
). Writing Yj = yj for 
and 
the joint posterior of X | Y is a generalized Dirichlet distribution with
where α'j = αj + yj and 
for 
.
[edit] See also
[edit] References
- ^ R. J. Connor and J. E. Mosiman 1969. Concepts of independence for proportions with a generalization of the Dirichlet distibution. Journal of the American Statistical Association, volume 64, pp194--206
- ^ T.-T. Wong 1998. Generalized Dirichlet distribution in Bayesian analysis. Applied Mathematics and Computation, volume 97, pp165-181
