An operational characterization of the notion of probability by algorithmic randomness and its applications

TL;DR

This paper introduces an operational way to understand probability using algorithmic randomness, specifically Martin-Loef randomness, and applies it to finite probability spaces, with implications for information theory and cryptography.

Contribution

It provides the first operational characterization of probability and independence via ensembles based on algorithmic randomness, extending to multiple events and applications.

Findings

Operational characterization of conditional probability using ensembles

Equivalent characterizations of independence for multiple events

Applications demonstrated in information theory and cryptography

Abstract

The notion of probability plays an important role in almost all areas of science and technology. In modern mathematics, however, probability theory means nothing other than measure theory, and the operational characterization of the notion of probability is not established yet. In this paper, based on the toolkit of algorithmic randomness we present an operational characterization of the notion of probability, called an ensemble. Algorithmic randomness, also known as algorithmic information theory, is a field of mathematics which enables us to consider the randomness of an individual infinite sequence. We use the notion of Martin-Loef randomness with respect to Bernoulli measure to present the operational characterization. As the first step of the research of this line, in this paper we consider the case of finite probability space, i.e., the case where the sample space of the underlying probability space is finite, for simplicity. We give a natural operational characterization of the notion of conditional probability in terms of ensemble, and give equivalent characterizations of the notion of independence between two events based on it. Furthermore, we give equivalent characterizations of the notion of independence of an arbitrary number of events/random variables in terms of ensembles. Moreover, we show that the independence of events/random variables is equivalent to the independence in the sense of van Lambalgen's Theorem, in the case where the underlying finite probability space is computable. In the paper we make applications of our framework to information theory, cryptography, and the simulation of a biased coin using fair coins, in order to demonstrate the wide applicability of our framework to the general areas of science and technology.