|
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.