  A new methodology is presented for the construction of control variates to
reduce the variance of additive functionals of Markov Chain Monte Carlo (MCMC)
samplers. Our control variates are definedthrough the minimization of the
asymptotic variance of the Langevin diffusion over a family of functions, which
can be seen as a quadratic risk minimization procedure. The use of these
control variates is theoretically justified. We show that the asymptotic
variances of some well-known MCMC algorithms, including the Random Walk
Metropolis and the (Metropolis) Unadjusted/Adjusted Langevin Algorithm, are
close to the asymptotic variance of the Langevin diffusion. Several examples of
Bayesian inference problems demonstrate that the corresponding reduction in the
variance is significant.
