Objective Bayesian Estimation for the Number of Classes in a Population Using Jeffreys and Reference Priors

Abstract

Estimation of the number of classes in a closed population is a problem that arises in many different subject areas. A common application occurs in animal populations where there is interest in determining the number of different species, also the diversity or species richness, of the population.

In this dissertation a class of models are considered, all with the common assumption that the number of individual items from each class, contributed to the sample, is a Poisson random variable. In order to conduct objective Bayesian inference, Jeffreys and reference priors are derived from the full likelihood. The Jeffreys and reference priors are functions of the information for the model parameters. The information is calculated in part using the linear difference score for integer parameter models (Lindsay & Roeder 1987).

A main accomplishment of this dissertation is deriving the form of the Jeffreys and reference priors for the number of classes. Both of the priors for the Jeffreys and reference methods factor into two independent priors, one for the parameter of interest and one for the nuisance parameters. This gives a justification for choosing these priors independent a priori, as is generally done for this problem. In this dissertation we prove that the posterior determined by these priors is proper for some low dimensional problems. With increased dimensionality, the form of the prior and posterior becomes increasingly intractable and we propose methods to deal with these difficulties. Microbial samples from the Framvaren Fjord (Behnke et al. 2006) and Lepidoptera data (Fisher et al. 1943) are used to illustrate that these priors can be used to implement a fully Bayesian procedure for the number of classes. The reference prior is comparable to maximum likelihood results, while the Jeffreys prior deviates from the frequentist estimates in models with larger numbers of parameters.