Exchangeability, the 'Histogram Theorem', and population inference

TL;DR

This paper introduces the Histogram Theorem for population inference under exchangeability, providing practical methods for predicting population outcomes from samples, with discussions on models and inference approaches.

Contribution

It presents the Histogram Theorem and its application to population inference under exchangeability, including non-parametric methods and a proof of the Representation Theorem.

Findings

Practical results for population inference under exchangeability.

Application of the Histogram Theorem to non-sampled population members.

Discussion of parametric vs. non-parametric models and marginalisation approaches.

Abstract

Some practical results are derived for population inference based on a sample, under the two qualitative conditions of 'ignorability' and exchangeability. These are the 'Histogram Theorem', for predicting the outcome of a non-sampled member of the population, and its application to inference about the population, both without and with groups. There are discussions of parametric versus non-parametric models, and different approaches to marginalisation. An Appendix gives a self-contained proof of the Representation Theorem for finite exchangeable sequences.