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

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.