hyperdirichlet-package {hyperdirichlet} | R Documentation

## The 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 <= 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.

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

p1^n1 * p2^n2 * p3^n3 * p4^n4 * p5^n5 * p6^n6.

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

p1^n1 * p2^n2 * p3^n3 * p4^n4 * p5^n5 * p6^n6 / (p1+p2+p3)^(r1+r2+r3)

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.

### Author(s)

Robin K. S. Hankin

### 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))

[Package hyperdirichlet version 1.3-8 Index]