Generalized Dirichlet distribution
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
.
The density function of
is
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
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.
, then
where
for 
and 
. Thus
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.
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 
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 puts a marble (sic) 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 puts the ball (sic) 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.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, 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
Neutral vector
In statistics, and specifically in the study of the Dirichlet distribution, a neutral vector of random variables is one that exhibits a particular type of statistical independence amongst its elements...
.
The density function of
iswhere 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
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.General moment function
If, thenwhere
for and . ThusReduction to standard Dirichlet distribution
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
Suppose is generalized Dirichlet, and that is multinomial with trials (here ). Writing for and the joint posterior of is a generalized Dirichlet distribution withwhere
and for Sampling experiment
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 puts a marble (sic) 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 puts the ball (sic) 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
- Multivariate Pólya distributionMultivariate Polya distributionThe multivariate Pólya distribution, named after George Pólya, also called the Dirichlet compound multinomial distribution, is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution with parameter vector \alpha, and a set of discrete samples is...
- Lukac's proportion-sum independence theorem