A cautionary note on the discrete uniform prior for the binomial N

Analysis of wildlife data frequently involves estimation of population size N based on binomial counts. Bayesian analysts often use a constant prior for N, the choice motivated by a desire to avoid an informative prior, and to let the data speak for themselves. For instance, data augmentation methods for model M-h posit a super-population of size M >> N with individual detection probabilities z(i)p(i), with p(i) sampled from a parametric family of interest, and z(i) an indicator of membership in the target population; thus, N = Sigma(i)z(i). Treating z(i) as independent Bernoulli trials with success rate and assigning a uniform prior to is equivalent to assigning a discrete uniform prior for N on {0,1,2, ... , M}; by setting M large enough, analysts approximate the improper constant prior on the positive integers. In this paper, I demonstrate some paradoxical and plainly unacceptable features of the constant prior. These defects are not shared by the scale prior, which has been recommended for its good performance as measured by frequentist criteria. I show how the scale prior can be approximated in program OpenBUGS, including data augmentation applications for individual covariates.