What is the plausibility of probability?(revised 2003, 2015)

TL;DR

This paper proves that rational uncertainty reasoning can be modeled using extended probability distributions, integrating mathematical results with philosophical and statistical arguments to support the robustness of Bayesian methods.

Contribution

It demonstrates that plausibility spaces of Cox's type can be embedded in ordered fields, implying that robust Bayesian analysis is a universal approach to uncertainty reasoning.

Findings

Plausibility spaces can be embedded in minimal ordered fields.

Robust Bayesian approach is shown to be universal.

Relation established between evidence theory and Bayesian analysis.

Abstract

We present and examine a result related to uncertainty reasoning, namely that a certain plausibility space of Cox's type can be uniquely embedded in a minimal ordered field. This, although a purely mathematical result, can be claimed to imply that every rational method to reason with uncertainty must be based on sets of extended probability distributions, where extended probability is standard probability extended with infinitesimals. This claim must be supported by some argumentation of non-mathematical type, however, since pure mathematics does not tell us anything about the world. We propose one such argumentation, and relate it to results from the literature of uncertainty and statistics. In an added retrospective section we discuss some developments in the area regarding countable additivity, partially ordered domains and robustness, and philosophical stances on the Cox/Jaynes approach since 2003. We also show that the most general partially ordered plausibility calculus embeddable in a ring can be represented as a set of extended probability distributions or, in algebraic terms, is a subdirect sum of ordered fields. In other words, the robust Bayesian approach is universal. This result is exemplified by relating Dempster-Shafer's evidence theory to robust Bayesian analysis.