Contextuality and dichotomizations of random variables

TL;DR

This paper justifies the use of dichotomizations of random variables within the Contextuality-by-Default framework, especially for categorical and real-valued variables, based on topological principles.

Contribution

It introduces a general principle for selecting dichotomizations of random variables using pre-topological structures, extending the approach to continuous and categorical data.

Findings

Dichotomizations split value spaces into linked subsets based on pre-topology.

Applicable to categorical variables by considering all dichotomizations.

For real-valued variables, dichotomizations are based on interval cuts.

Abstract

The Contextuality-by-Default approach to determining and measuring the (non)contextuality of a system of random variables requires that every random variable in the system be represented by an equivalent set of dichotomous random variables. In this paper we present general principles that justify the use of dichotomizations and determine their choice. The main idea in choosing dichotomizations is that if the set of possible values of a random variable is endowed with a pre-topology (V-space), then the allowable dichotomizations split the space of possible values into two linked subsets ("linkednes" being a weak form of pre-topological connectedness). We primarily focus on two types of random variables most often encountered in practice: categorical and real-valued ones (including continuous random variables, greatly underrepresented in the contextuality literature). A categorical variable (one with a finite number of unordered values) is represented by all of its possible dichotomizations. If the values of a random variable are real numbers, then they are dichotomized by intervals above and below a variable cut point.