ll{
Package: hyperdirichlet
Type: Package
Version: 1.1-8
Date: 2008-03-26
License: GPL
}This package provides a generalization of the Dirichlet distribution
that is useful for analyzing multinomial trials with a priori
restrictions.
As an example, consider six people (players), numbered 1 to 6.
These players are members of a running club and regularly race one
another.
Each player has an associated number $p_1$ to $p_6$, with
$0\leq p_i\leq 1$ for $i=1,\ldots,6$ and
$\sum_{i=1}^6 p_i=1$. If all six take part in a
race, then the probability that player $i$ wins is simply $p_i$.
We wish to make inferences about the $p_i$ from their performances.
If all six race and $p_i$ wins $n_i$, then the likelihood
function is just
$${p_1}^{n_1}\cdot{p_2}^{n_2}\cdot{p_3}^{n_3}\cdot{p_4}^{n_4}\cdot{p_5}^{n_5}\cdot{p_6}^{n_6}.$$
With a uniform prior, the posterior is Dirichlet.
The players now have a race but only $p_1$, $p_2$ and $p_3$
take place, winning $r_1$, $r_2$ and $r_3$ respectively.
The likelihood function is then
$$\frac{
{p_1}^{n_1+r_1}\cdot{p_2}^{n_2+r_2}\cdot{p_3}^{n_3+r_3}\cdot{p_4}^{n_4}\cdot{p_5}^{n_5}\cdot{p_6}^{n_6}
}{
\left(p_1+p_2+p_3\right)^{r_1+r_2+r_3
}
}$$
This distribution is not a Dirichlet distribution but is representable
in this package; the R idiom would be
jj <- dirichlet(powers = c(5,4,3,5,3,2))
jj <- jj + mult_restricted_obs(6, 1:3, c(4,5,2))
where the first line specifies a Dirichlet distribution for the all-play
data and the second line augments the likelihood with the observations
from the restricted race.