$L_2$-norm posterior contraction in Gaussian models with unknown variance

TL;DR

This paper develops a new testing approach for establishing posterior contraction rates in Gaussian models with unknown variance, extending existing methods to more realistic settings.

Contribution

It introduces a novel test function suitable for unknown variance scenarios and provides conditions for posterior contraction in high-dimensional and nonparametric regression.

Findings

Derived a test function for unknown variance Gaussian models

Established posterior contraction rates under new conditions

Applied results to high-dimensional and nonparametric regression

Abstract

The testing-based approach is a fundamental tool for establishing posterior contraction rates. Although the Hellinger metric is attractive owing to the existence of a desirable test function, it is not directly applicable in Gaussian models, because translating the Hellinger metric into more intuitive metrics typically requires strong boundedness conditions. When the variance is known, this issue can be addressed by directly constructing a test function relative to the $L_2$-metric using the likelihood ratio test. However, when the variance is unknown, existing results are limited and rely on restrictive assumptions. To overcome this limitation, we derive a test function tailored to an unknown variance setting with respect to the $L_2$-metric and provide sufficient conditions for posterior contraction based on the testing-based approach. We apply this result to analyze high-dimensional regression and nonparametric regression.