Computable Random Variables and Conditioning

TL;DR

This paper develops an elementary, computable framework for random variables and conditioning, grounded in valuations and type-two effectivity, linking probability theory with Turing computation.

Contribution

It introduces a computable theory of conditional random variables within a type-two effectivity framework, extending prior work with a mathematically explicit approach.

Findings

Provides a computable model of random variables and conditioning.

Establishes a link between probability theory and Turing computation.

Uses lower-measurable sets to control limits of open sets.

Abstract

The aim of this paper is to present an elementary computable theory of random variables, based on the approach to probability via valuations. The theory is based on a type of lower-measurable sets, which are controlled limits of open sets, and extends existing work in this area by providing a computable theory of conditional random variables. The theory is based within the framework of type-two effectivity, so has an explicit direct link with Turing computation, and is expressed in a system of computable types and operations, so has a clean mathematical description.