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

Simple variance bounds with applications to Bayesian posteriors and intractable distributions