Weakly informative reparameterisations for location-scale mixtures

TL;DR

This paper introduces a new parameterisation and weakly informative priors for univariate Gaussian mixture models, enabling proper posteriors and exchangeability in Bayesian analysis, with an R package implementation.

Contribution

It develops a novel reparameterisation and prior for Gaussian mixtures that ensures proper posteriors and exchangeability, addressing longstanding Bayesian analysis challenges.

Findings

Proper posteriors with minimal sample sizes

Exchangeability of MCMC samples demonstrated

Implementation available in R package Ultimixt

Abstract

While mixtures of Gaussian distributions have been studied for more than a century (Pearson, 1894), the construction of a reference Bayesian analysis of those models still remains unsolved, with a general prohibition of the usage of improper priors (Fruwirth-Schnatter, 2006) due to the ill-posed nature of such statistical objects. This difficulty is usually bypassed by an empirical Bayes resolution (Richardson and Green, 1997). By creating a new parameterisation cantered on the mean and possibly the variance of the mixture distribution itself, we manage to develop here a weakly informative prior for a wide class of mixtures with an arbitrary number of components. We demonstrate that some posterior distributions associated with this prior and a minimal sample size are proper. We provide MCMC implementations that exhibit the expected exchangeability. We only study here the univariate case, the extension to multivariate location-scale mixtures being currently under study. An R package called Ultimixt is associated with this paper.