The work presented investigates speeding up MCMC methods by introducing control variates
based on approximate solutions of the Poisson equation. In the setting of Metropolis-Hastings
chains in Rd two scalable approaches of approximately solving the Poisson equation are discussed.
In both cases an underlying weakly convergent sequence of related Markov chains,
enumerated by a scaling parameter, is identi ed and results, asymptotic in the scaling parameter,
are given for the achieved improvement.
In the rst approach control variates are constructed according to a sequence of ner and
ner partitions of the state-space of the Metropolis-Hastings chain, with the mesh of the partition
serving as the scaling parameter. In this context it is shown, that as the mesh reduces
arbitrarily, so does the asymptotic variance in the Central limit theorem associated with the
control variate given by the partition.
The second approach assumes a target density of a product type and scales the dimension
of the state-space and the variance of the proposal simultaneously. The resulting weakly
convergent sequence converges to a Langevin di usion, which is then used to construct control
variates for the Metropolis-Hastings chains in the sequence. The bounds obtained in this
context suggest the improvement achieved by this approach grows almost linearly in dimension.