  We introduce a general framework that constructs estimators with reduced
variance for random walk Metropolis and Metropolis-adjusted Langevin
algorithms. The resulting estimators require negligible computational cost and
are derived in a post-process manner utilising all proposal values of the
Metropolis algorithms. Variance reduction is achieved by producing control
variates through the approximate solution of the Poisson equation associated
with the target density of the Markov chain. The proposed method is based on
approximating the target density with a Gaussian and then utilising accurate
solutions of the Poisson equation for the Gaussian case. This leads to an
estimator that uses two key elements: (i) a control variate from the Poisson
equation that contains an intractable expectation under the proposal
distribution, (ii) a second control variate to reduce the variance of a Monte
Carlo estimate of this latter intractable expectation. Simulated data examples
are used to illustrate the impressive variance reduction achieved in the
Gaussian target case and the corresponding effect when target Gaussianity
assumption is violated. Real data examples on Bayesian logistic regression and
stochastic volatility models verify that considerable variance reduction is
achieved with negligible extra computational cost.
