Applications of the Beta Distribution Part 1: Transformation Group Approach

TL;DR

This paper introduces a transformation group approach to the beta distribution prior, linking it with Poisson and gamma distributions, and explores its applications in classification, prediction, and genetics.

Contribution

It proposes a novel transformation group method for the beta distribution prior, enhancing modeling of finite data sets and clarifying its relationship with other distributions.

Findings

The approach accounts for finite data sets by limiting parameter ranges.

It elucidates the connection between beta, Poisson, and gamma distributions.

Applications in genetics demonstrate the method's practical utility.

Abstract

A transformation group approach to the prior for the parameters of the beta distribution is suggested which accounts for finite sets of data by imposing a limit to the range of parameter values under consideration. The relationship between the beta distribution and the Poisson and gamma distributions in the continuum is explored, with an emphasis on the decomposition of the model into separate estimates for size and shape. Use of the beta distribution in classification and prediction problems is discussed, and the effect of the prior on the analysis of some well known examples from statistical genetics is examined.