Tails assumptions and posterior concentration rates for mixtures of Gaussians

TL;DR

This paper investigates how tail assumptions affect posterior convergence rates in Gaussian mixture models, revealing that location mixtures often outperform location-scale mixtures under heavy tails, and proposing hybrid models for improved performance.

Contribution

The study challenges existing beliefs by showing location mixtures outperform location-scale mixtures under heavy tails and introduces hybrid models that outperform both, regardless of tail assumptions.

Findings

Location mixtures outperform location-scale mixtures under heavy tails.

Hybrid location-scale mixtures outperform both traditional types.

Releasing tail assumptions requires covariate-dependent priors.

Abstract

Nowadays in density estimation, posterior rates of convergence for location and location-scale mixtures of Gaussians are only known under light-tail assumptions; with better rates achieved by location mixtures. It is conjectured, but not proved, that the situation should be reversed under heavy tails assumptions. The conjecture is based on the feeling that there is no need to achieve a good order of approximation in regions with few data (say, in the tails), favoring location-scale mixtures which allow for spatially varying order of approximation. Here we test the previous argument on the Gaussian errors mean regression model with random design, for which the light tail assumption is not required for proofs. Although we cannot invalidate the conjecture due to the lack of lower bound, we find that even with heavy tails assumptions, location-scale mixtures apparently perform always worst than location mixtures. However, the proofs suggest to introduce hybrid location-scale mixtures that are find to outperform both location and location-scale mixtures, whatever the nature of the tails. Finally, we show that all tails assumptions can be released at the price of making the prior distribution covariate dependent.