Diffuse Models for Sampling and Predictive Inference

Abstract

As a natural, intuitive model for inferences about certain characteristics of finite populations, Bruce Hill has proposed a sequence of exchangeable variables $X_1, \cdots, X_{n + 1}$ which have distinct values with probability one and have the property that, conditional on $X_1, \cdots, X_n$, the next observation $X_{n + 1}$ is equally likely to fall in any of the $n + 1$ intervals determined by $X_1, \cdots, X_n$. Harold Jeffreys had previously assumed such a model (in the case $n = 2$) for normal measurements with unknown mean and variance. Hill has shown that, for $n \geqslant 1$, there exist no countably additive distributions with the prescribed properties. It is shown here that finitely additive distributions with these properties do exist for all $n$ and have a number of interesting properties.