ewens

The R package ewens gives functions for the Ewens sampling distribution. The Ewens distribution is a probability distribution over partitions of an integer \(n\). It is governed by a single non-negative parameter, \(\theta\). The larger the value of \(\theta\), the more parts there are in the partition. If \(\theta\) is equal to zero, there is just a single partition with size \(n\).

The Ewens distribution can be thought of as a distribution over partitions of an integer, but it originated in population genetics to give the probability of alleles in a sample, and sampling from the Ewens distribution involves running through the Chinese Restaurant Process \(n\) times. For further information on the Ewens sampling formula, see Simon Tavaré’s The magical Ewens sampling formula or Harry Crane’s The Ubiquitous Ewens Sampling Formula.

Installation

You can install the development version of ewens like so:

if (!requireNamespace("remotes", quietly = TRUE)) {
  install.packages("remotes")
}
remotes::install_github("chrishanretty/ewens")

Drawing from the Ewens distribution

Suppose I wish to divide a class of 24 into groups of different sizes. I set \(\theta\) to one, and call rewens

set.seed(2923)
library(ewens)

allocation <- rewens(24, theta = 1)

The vector allocation gives the groups to which each student has been assigned. The sizes of the groups can be recovered by tabulating the result.

table(allocation)

The probability mass function of the Ewens distribution

Suppose a student challenges their allocation into a particular group. What can we say about the probability of the particular allocation, given \(\theta = 1\)?

pr <- dewens(allocation, theta = 1)
pr

The probability of drawing this particular allocation is very low – but then again, given that there are 1575 different partitions of twenty-four, we should not be too surprised.

The number of parts in the partition

It is also possible to calculate the probability that there are \(x\) parts in a partition of \(n\) given \(\theta\). This is done using dewens_k. Thus, the probability of creating the groups above is given by

dewens_k(k = length(unique(allocation)),
         n = 24,
         theta = 1)