Hierarchical Bayesian Estimation for the Number of Species Josemar Rodrigues, Luis A. Milan, and Jose ´ G. Leite DEs-UFSCar-CP-676-13565-905-Sa ˜o Carlos-Brazil Summary This paper is concerned with the estimation of the number of species in a population through a fully hierarchical Bayesian model using the Metropolis algorithm. The proposed Bayesian estimator is based on Poisson random variables with means that are distributed according to some prior distributions with unknown hyperparameters. An empirical Bayes approach is considered and compared with the fully Bayesian approach based on biological data. Key words: Empirical Bayes; Metropolis algorithm; Poisson-Gamma mixture distribution. 1. Introduction We propose a method for estimating the number N of species in a population through an exact Bayesian hierarchical model. This method is a fully Bayesian development of an empirical approach proposed by Bunge and Freeman-Gallant (1996), which consider the abundance of each species contributing in a sample as a Poisson random variable with mean q i , i ¼ 1; ... ; N. They consider N as an unknown fixed parameter and assign a prior distribution for q i with unknown hyperpameters which are estimated from the data. The properties of these estima- tors are based on the asymptotic results. They also suggest the hierarchical Bayes model approach. Bunge and Fitzpatrick (1993) provides an interesting review of the problem of estimating the number of species and a more general model can be found in Leite et al. (2000). In order to formulate the problem consider the following notations: X i is the number of individuals from species i in the sample (unobservable), i ¼ 1; ... ; N; N is the number of species in the population (unknown); n j ¼ P N i¼1 I ðX i ¼jÞ ; j  1 are the frequencies of frequencies where I A means the indicator function of the set A; n ¼ P j1 jn j is the sample size and w ¼ P j1 n j the number of different species in the sample. We implement a fully Bayesian hierarchical procedure for a special prior distri- bution in Section 2. In Section 3 we introduce the empirical Bayes approach. An Biometrical Journal 43 (2001) 6, 737–746 # WILEY-VCH Verlag Berlin GmbH, 13086 Berlin, 2001 0323-3847/01/0610-0737 $17.50þ.50/0