  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.
