Systems of Precision: Coherent Probabilities on Pre-Dynkin-Systems and Coherent Previsions on Linear Subspaces

TL;DR

This paper explores the structure of imprecise probabilities on specific set systems called pre-Dynkin-systems, establishing their mathematical properties, dualities, and extensions to expectation frameworks, with implications across machine learning, quantum probability, and decision theory.

Contribution

It demonstrates that systems of precision form pre-Dynkin-systems and establishes their embedding into algebras, revealing coherence and extendability equivalences and extending the framework to linear subspaces and partial expectations.

Findings

Pre-Dynkin-systems can be embedded into algebras of sets under extendability.

Coherence and extendability are equivalent conditions in this framework.

A lattice duality relates systems of precision to credal sets of probabilities.

Abstract

In literature on imprecise probability little attention is paid to the fact that imprecise probabilities are precise on a set of events. We call these sets systems of precision. We show that, under mild assumptions, the system of precision of a lower and upper probability form a so-called (pre-)Dynkin-system. Interestingly, there are several settings, ranging from machine learning on partial data over frequential probability theory to quantum probability theory and decision making under uncertainty, in which a priori the probabilities are only desired to be precise on a specific underlying set system. Here, (pre-)Dynkin-systems have been adopted as systems of precision, too. We show that, under extendability conditions, those pre-Dynkin-systems equipped with probabilities can be embedded into algebras of sets. Surprisingly, the extendability conditions elaborated in a strand of work in quantum probability are equivalent to coherence from the imprecise probability literature. On this basis, we spell out a lattice duality which relates systems of precision to credal sets of probabilities. We conclude the presentation with a generalization of the framework to expectation-type counterparts of imprecise probabilities. The analogue of pre-Dynkin-systems turn out to be (sets of) linear subspaces in the space of bounded, real-valued functions. We introduce partial expectations, natural generalizations of probabilities defined on pre-Dynkin-systems. Again, coherence and extendability are equivalent. A related, but more general lattice duality preserves the relation between systems of precision and credal sets of probabilities.