From First Principles to Multi-scale Decomposition:Mutual Information as a Segregation Index

TL;DR

This paper derives a multi-scale segregation index based on mutual information, providing a principled way to measure and analyze demographic segregation across geographic and demographic scales.

Contribution

It uniquely characterizes mutual information as a segregation index through first principles, enabling detailed multi-scale analysis and insights into ethnic segregation.

Findings

Mutual information satisfies key properties for segregation measurement.

Decomposition of mutual information reveals scale-specific segregation contributions.

Application to England and Wales uncovers insights inaccessible to other indices.

Abstract

Segregation is a multi-scale phenomenon that requires careful measurement. A segregation index implicitly defines how the demographic compositions of locations are compared. We identify two properties -- mean-minimisation and invariance -- that uniquely characterise the Kullback-Leibler divergence as a measure of demographic difference. Mean-minimiser makes the comparison consistent with population aggregation and invariance ensures that it behaves intuitively under demographic coarse-graining. The corresponding segregation index is mutual information, which can be decomposed across geographic and demographic scales to identify the contributions of regions and supergroups. We demonstrate how this reveals insights into ethnic residential segregation in England and Wales that would be inaccessible otherwise. By deriving mutual information from first principles, we identify situations in which it is the only suitable segregation index, and provide open source software to support multi-scale analysis.