Some simple variance bounds from Stein’s method

Using coupling techniques based on Stein’s method for probability approximation, we revisit
classical variance bounding inequalities of Chernoff, Cacoullos, Chen and Klaassen. Our bounds are
immediate in any context wherein a Stein identity is available. After providing illustrative examples
for a Gaussian and a Gumbel target distribution, our main contributions are new variance bounds in settings where the underlying density function is unknown or intractable. Applications include bounds for analysis of the posterior in Bayesian statistics, bounds for asymptotically Gaussian random variables using zero-biased couplings, and bounds for random variables which are New Better (Worse) than Used in Expectation.