Asymptotic Consistency of Loss-Calibrated Variational Bayes

TL;DR

This paper proves the asymptotic consistency of loss-calibrated variational Bayes (LCVB) and decision rules derived from it, ensuring reliable Bayesian inference and decision-making in large-sample settings.

Contribution

It establishes the asymptotic consistency of LCVB and decision rules, extending theoretical guarantees for loss-aware Bayesian methods.

Findings

Asymptotic consistency of the calibrated approximate posterior

Asymptotic consistency of decision rules from LCVB

Consistency of naive variational Bayesian decision rules

Abstract

This paper establishes the asymptotic consistency of the {\it loss-calibrated variational Bayes} (LCVB) method. LCVB was proposed in~\cite{LaSiGh2011} as a method for approximately computing Bayesian posteriors in a `loss aware' manner. This methodology is also highly relevant in general data-driven decision-making contexts. Here, we not only establish the asymptotic consistency of the calibrated approximate posterior, but also the asymptotic consistency of decision rules. We also establish the asymptotic consistency of decision rules obtained from a `naive' variational Bayesian procedure.