hyperdirichlet-package: The Hyperdirichlet package

In hyperdirichlet: A Generalization of the Dirichlet Distribution

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

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))

hyperdirichlet documentation built on May 31, 2017, 5:18 a.m.