Predictive distributions that mimic frequencies over a restricted subdomain (expanded preprint version)

TL;DR

This paper explores predictive distributions that mimic observed frequencies over a limited subdomain, providing theoretical insights, computational tools, and new theorems to understand their properties and limitations in finite inference contexts.

Contribution

It introduces a framework for frequency mimicking over finite domains, develops computational software, and establishes new theorems clarifying their applicability and limitations.

Findings

Provides a software tool for generating frequency mimicking distributions.

Identifies the structure of adherent masses in finitely additive distributions.

Shows limitations of extending frequency mimicking assertions to larger N.

Abstract

A predictive distribution over a sequence of $N+1$ events is said to be "frequency mimicking" whenever the probability for the final event conditioned on the outcome of the first $N$ events equals the relative frequency of successes among them. Infinitely extendible exchangeable distributions that universally inhere this property are known to have several annoying concomitant properties. We motivate frequency mimicking assertions over a limited subdomain in practical problems of finite inference, and we identify their computable coherent implications. We provide some computed examples using reference distributions, and we introduce computational software to generate any specification. The software derives from an inversion of the finite form of the exchangeability representation theorem. Three new theorems delineate the extent of the usefulness of such distributions, and we show why it may not be appropriate to extend the frequency mimicking assertions for a specified value of $N$ to any arbitrary larger size of $N$. The constructive results identify the source and structure of "adherent masses" in the limit of a sequence of finitely additive distributions. Appendices develop a novel geometrical representation of conditional probabilities which illuminate the analysis.