Nonasymptotic Laplace approximation under model misspecification

TL;DR

This paper develops non-asymptotic bounds for the log-marginal likelihood in Bayesian inference, accommodating model misspecification and high-dimensional parameters without relying on asymptotic posterior shapes.

Contribution

It introduces non-asymptotic two-sided bounds for the log-marginal likelihood that include the classical Laplace approximation as a special case, even under model misspecification.

Findings

Provides bounds valid for growing parameter dimensions

Allows for model misspecification in Bayesian analysis

Does not assume asymptotic posterior shape

Abstract

We present non-asymptotic two-sided bounds to the log-marginal likelihood in Bayesian inference. The classical Laplace approximation is recovered as the leading term. Our derivation permits model misspecification and allows the parameter dimension to grow with the sample size. We do not make any assumptions about the asymptotic shape of the posterior, and instead require certain regularity conditions on the likelihood ratio and that the posterior to be sufficiently concentrated.