Valid Credible Ellipsoids for Linear Functionals by a Renormalized Bernstein-von Mises Theorem

TL;DR

This paper establishes a renormalized Bernstein-von Mises theorem in infinite-dimensional Gaussian models, ensuring the validity of credible sets for linear functionals without requiring traditional solution conditions.

Contribution

It introduces a renormalized BvM theorem that guarantees credible ellipsoids are valid confidence sets in high-dimensional Gaussian regression models.

Findings

Credible ellipsoids serve as asymptotic confidence sets.

The method applies to problems where the information equation has no solution.

It recovers semi-parametric BvM results for Schrödinger problems.

Abstract

We consider an infinite-dimensional Gaussian regression model, equipped with a high-dimensional Gaussian prior. We address the frequentist validity of posterior credible sets for a vector of linear functionals. We specify conditions for a 'renormalized' Bernstein-von Mises theorem (BvM), where the posterior, centered at its mean, and the posterior mean, centered at the ground truth, have the same normal approximation. This requires neither a solution to the information equation nor a $\sqrt{N}$-consistent estimator. We show that our renormalized BvM implies that a credible ellipsoid, specified by the mean and variance of the posterior, is an asymptotic confidence set. For a single linear functional, we identify a credible ellipsoid with a symmetric credible interval around the posterior mean. We bound the diameter. We check our conditions for Darcy's problem, where the information equation has no solution in natural settings. For the Schr\"odinger problem, we recover an efficient semi-parametric BvM from our renormalized BvM.