Popper Functions, Lexicographical Probability, and Non-Archimedean Probability

TL;DR

This paper explores alternative probability theories—Popper functions, non-Archimedean, and lexicographic probabilities—showing their interrelations and how they can model scenarios standard probability cannot handle well.

Contribution

It establishes a representation theorem linking Popper functions and non-Archimedean probabilities and demonstrates their equivalence and interchangeability with lexicographic probabilities.

Findings

Every non-Archimedean probability is infinitesimally close to a Popper function.

Non-Archimedean probabilities can be represented lexicographically.

Popper functions, non-Archimedean, and lexicographic probabilities are essentially interchangeable.

Abstract

Standard probability theory has been extremely successful but there are some conceptually possible scenarios, such as fair infinite lotteries, that it does not model well. For this reason alternative probability theories have been formulated. We look at three of these: Popper functions, a specific kind of non-Archimedean probability functions, and lexicographic probability functions. We relate Popper functions to non-Archimedean probability functions (of a specific kind) by means of a representation theorem: every non-Archimedean probability function is infinitesimally close to some Popper function, and vice versa. We also show that non-Archimedean probability functions can be given a lexicographic representation. Thus Popper functions, a specific kind of non-Archimedean probability functions, and lexicographic probability functions triangulate to the same place: they are in a good sense interchangeable.