Simple variance bounds with applications to Bayesian posteriors and intractable distributions

TL;DR

This paper introduces new variance bounds using coupling techniques based on Stein's method, applicable to intractable distributions and Bayesian posteriors, improving classical inequalities with modern probabilistic tools.

Contribution

It develops novel variance bounds leveraging modern coupling methods, extending classical inequalities to complex, intractable, or unknown density scenarios.

Findings

New variance bounds for asymptotically Gaussian variables

Bounds for New Better/Worse Than Used in Expectation variables

Analysis of Bayesian posterior distributions

Abstract

Using coupling techniques based on Stein's method for probability approximation, we revisit classical variance bounding inequalities of Chernoff, Cacoullos, Chen and Klaassen. Taking advantage of modern coupling techniques allows us to establish novel variance bounds in settings where the underlying density function is unknown or intractable. Applications include bounds for asymptotically Gaussian random variables using zero-biased couplings, bounds for random variables which are New Better (Worse) than Used in Expectation, and analysis of the posterior in Bayesian statistics.