hyperdirichlet (version 1.5-1)

hyperdirichlet-package:

Description

A generalization of the Dirichlet distribution

Details

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 $0\le p_i\le 1$ for $i=1,\dots ,6$ 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

$$\frac{{p_1}^{n_1+r_1}\cdot {p_2}^{n_2+r_2}\cdot {p_3}^{n_3+r_3}\cdot {p_4}^{n_4}\cdot {p_5}^{n_5}\cdot {p_6}^{n_6}}{{(p_1+p_2+p_3)}^{r_1+r_2+r_3}}$$

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.

References

Robin K. S. Hankin (2010). “A Generalization of the Dirichlet Distribution”, Journal of Statistical Software, 33(11), 1-18, http://www.jstatsoft.org/v33/i11/

Examples

  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))