Natural induction: An objective bayesian approach

Abstract

The statistical analysis of a sample taken from a finite population is a classic problem for which no generally accepted objective Bayesian results seem to exist. Bayesian solutions to this problem may be very sensitive to the choice of the prior, and there is no consensus as to the appropriate prior to use.This paper uses new developments in reference prior theory to justify and generalize Perks (1947) ([15]) ‘rule of succession’ — determining the probability that a new element from a population will have a property, given that all n previous elements from a random sample possessed the property — and to propose a new objective Bayesian solution to the ‘law of natural induction’ problem — determining the probability that all elements in a finite population have the property, given that all previous elements had the property.The prior used for the first problem is the reference prior for an underlying hypergeometric probability model, a prior first suggested by Jeffreys (1946) ([10]) and recently justified on the basis of an exchangeability argument in Berger, Bernardo and Sun (2009) ([4]). The reference prior in the second problem arises as a modification to this prior that results from declaring the quantity of interest to be whether or not all the elements in the finite population have the property under scrutiny.