Generalized Dirichlet distribution

Summary

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 vectors with a GD distribution are completely neutral.[1]

The density function of is

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

For example, if k=4, then the density function of is

where and .

Connor and Mosimann define the PDF as they did for the following reason. Define random variables with . Then have the generalized Dirichlet distribution as parametrized above, if the are independent beta with parameters , .

Alternative form given by Wong edit

Wong[2] gives the slightly more concise form for  

 

where   for   and  . Note that Wong defines a distribution over a   dimensional space (implicitly defining  ) while Connor and Mosiman use a   dimensional space with  .

General moment function edit

If  , then

 

where   for   and  . Thus

 

Reduction to standard Dirichlet distribution edit

As stated above, if   for   then the distribution reduces to a standard Dirichlet. This condition is different from the usual case, in which setting the additional parameters of the generalized distribution to zero results in the original distribution. However, in the case of the GDD, this results in a very complicated density function.

Bayesian analysis edit

Suppose   is generalized Dirichlet, and that   is multinomial with   trials (here  ). Writing   for   and   the joint posterior of   is a generalized Dirichlet distribution with

 

where   and   for  

Sampling experiment edit

Wong gives the following system as an example of how the Dirichlet and generalized Dirichlet distributions differ. He posits that a large urn contains balls of   different colours. The proportion of each colour is unknown. Write   for the proportion of the balls with colour   in the urn.

Experiment 1. Analyst 1 believes that   (ie,   is Dirichlet with parameters  ). The analyst then makes   glass boxes and puts   marbles of colour   in box   (it is assumed that the   are integers  ). Then analyst 1 draws a ball from the urn, observes its colour (say colour  ) and puts it in box  . He can identify the correct box because they are transparent and the colours of the marbles within are visible. The process continues until   balls have been drawn. The posterior distribution is then Dirichlet with parameters being the number of marbles in each box.

Experiment 2. Analyst 2 believes that   follows a generalized Dirichlet distribution:  . All parameters are again assumed to be positive integers. The analyst makes   wooden boxes. The boxes have two areas: one for balls and one for marbles. The balls are coloured but the marbles are not coloured. Then for  , he puts   balls of colour  , and   marbles, in to box  . He then puts a ball of colour   in box  . The analyst then draws a ball from the urn. Because the boxes are wood, the analyst cannot tell which box to put the ball in (as he could in experiment 1 above); he also has a poor memory and cannot remember which box contains which colour balls. He has to discover which box is the correct one to put the ball in. He does this by opening box 1 and comparing the balls in it to the drawn ball. If the colours differ, the box is the wrong one. The analyst places a marble in box 1 and proceeds to box 2. He repeats the process until the balls in the box match the drawn ball, at which point he places the ball in the box with the other balls of matching colour. The analyst then draws another ball from the urn and repeats until   balls are drawn. The posterior is then generalized Dirichlet with parameters   being the number of balls, and   the number of marbles, in each box.

Note that in experiment 2, changing the order of the boxes has a non-trivial effect, unlike experiment 1.

See also edit

References edit

  1. ^ R. J. Connor and J. E. Mosiman 1969. Concepts of independence for proportions with a generalization of the Dirichlet distribution. Journal of the American Statistical Association, volume 64, pp. 194–206
  2. ^ T.-T. Wong 1998. Generalized Dirichlet distribution in Bayesian analysis. Applied Mathematics and Computation, volume 97, pp. 165–181