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.
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))
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