| 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 $0 <= p_i="" <="1$" for="" $1,...,6$="" and="" $p_1="" +="" ...="" p_6="1$." if="" all="" six="" take="" part="" in="" a="" race,="" then="" the="" probability="" that="" player="" $i$="" wins="" is="" simply="" $p_i$.<="" p="">
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.
jj <- dirichlet(powers = c(5,4,3,5,3,2))
jj <- jj + mult_restricted_obs(6, 1:3, c(4,5,2))
data(icons)
maximum_likelihood(as.hyperdirichlet(icons))
Run the code above in your browser using DataLab