Dirichlet process mixtures under affine transformations of the data

TL;DR

This paper enhances Dirichlet process Gaussian mixture models to be invariant under affine transformations, ensuring robustness in density estimation and clustering, validated through simulations and astronomical data analysis.

Contribution

It introduces a prior specification that guarantees affine invariance and formalizes asymptotic robustness of DPM-G models under data transformations.

Findings

Model invariance under affine transformations achieved

Asymptotic robustness established under mild conditions

Validated through simulations and real astronomical data

Abstract

Location-scale Dirichlet process mixtures of Gaussians (DPM-G) have proved extremely useful in dealing with density estimation and clustering problems in a wide range of domains. Motivated by an astronomical application, in this work we address the robustness of DPM-G models to affine transformations of the data, a natural requirement for any sensible statistical method for density estimation and clustering. First, we devise a coherent prior specification of the model which makes posterior inference invariant with respect to affine transformations of the data. Second, we formalise the notion of asymptotic robustness under data transformation and show that mild assumptions on the true data generating process are sufficient to ensure that DPM-G models feature such a property. Our investigation is supported by an extensive simulation study and illustrated by the analysis of an astronomical dataset consisting of physical measurements of stars in the field of the globular cluster NGC 2419.