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.
This reduces to the standard Dirichlet distribution if
Connor and Mosimann define the PDF as they did for the following reason.
Define random variables
have the generalized Dirichlet distribution as parametrized above, if the
are independent beta with parameters
Wong[2] gives the slightly more concise form for
Note that Wong defines a distribution over a
dimensional space (implicitly defining
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.
is a generalized Dirichlet distribution with where
He posits that a large urn contains balls of
for the proportion of the balls with colour
glass boxes and puts
He can identify the correct box because they are transparent and the colours of the marbles within are visible.
The posterior distribution is then Dirichlet with parameters being the number of marbles in each box.
follows a generalized Dirichlet distribution:
All parameters are again assumed to be positive integers.
The boxes have two areas: one for balls and one for marbles.
He then puts a ball of colour
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.
If the colours differ, the box is the wrong one.
The analyst then draws another ball from the urn and repeats until
The posterior is then generalized Dirichlet with parameters
Note that in experiment 2, changing the order of the boxes has a non-trivial effect, unlike experiment 1.