Thank you very much to Jim Garrett, Patrick Bélisle and David
Spiegelhalter for their answers. I'll resume their suggestions.
First, it is not possible to have uncertainty on the parameters of the
Dirichlet distribution and learn about them. It's just not a facility
within BUGS at the moment and it is not a priority.
1/ first suggestion :
The Dirichlet distribution is the distribution of independent,
equally-scaled gamma variates divided by their sum, perhaps you could
work with gamma variables instead. For example, if
z_1 ~ gamma(alpha_1, 1),
z_2 ~ gamma(alpha_2, 1), and
z_3 ~ gamma(alpha_3, 1),
and they are independent, and
K = z_1 + z_2 + z_3,
then z_1/K, z_2/K, and z_3/K are jointly Dirichlet.
2/ second suggestion : logistic-normal models.
Having said this, I read an excellent book on compositional data
"Analysis of Compositional Data" by Aitchison, from Chapman and Hall
publishers.
The author points out several limitations to the Dirichlet stemming
from the independence assumption, and recommends an alternative in
which P variables that sum to one are transformed to P-1 variables
that are unconstrained, so that the (P-1)-vector may reasonably be
modeled by a multivariate normal. This allows for correlations among
the proportions in addition to the correlations implied by the
unit-sum constraint. If I recall correctly, the author refers to the
generalized logistic transformation, which would be (following the
example above)
s_1 = log(z_1 / z_3)
s_2 = log(z_2 / z_3)
s_3 = 0 (i.e., = log(z_3 / z_3) )
The choice of which component will be in the denominator is completely
arbitrary.
This transformation has the inverse
z_1 = exp(s_1) / (exp(s_1) + exp(s_2) + exp(s_3) ),
z_2 = exp(s_2) / (exp(s_1) + exp(s_2) + exp(s_3) ), and
z_3 = exp(s_3) / (exp(s_1) + exp(s_2) + exp(s_3) ).
You could then suppose that (s_1, s_2) ~ bivariate normal and estimate
both the mean and variance matrix. It seems to me likely to be
straightforward. It is a richer model for the data. (but it cannot
handle zero compositions).
Claire Chabanet
INRA - Laboratoire de Recherches sur les Aromes
17 rue SULLY, BV1540
21034 DIJON cedex FRANCE
e-mail : [log in to unmask]
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