Source: J. Appl. Probab. Volume 46, Number 4
(2009), 938-959.
The waste-recycling Monte Carlo (WRMC) algorithm introduced by physicists
is a modification of the (multi-proposal) Metropolis--Hastings algorithm,
which makes use of all the proposals in the empirical mean, whereas the
standard (multi-proposal) Metropolis--Hastings algorithm uses only the
accepted proposals. In this paper we extend the WRMC algorithm to a
general control variate technique and exhibit the optimal choice of the
control variate in terms of the asymptotic variance. We also give an
example which shows that, in contradiction to the intuition of physicists,
the WRMC algorithm can have an asymptotic variance larger than that of the
Metropolis--Hastings algorithm. However, in the particular case of the
Metropolis--Hastings algorithm called the Boltzmann algorithm, we prove
that the WRMC algorithm is asymptotically better than the
Metropolis--Hastings algorithm. This last property is also true for the
multi-proposal Metropolis--Hastings algorithm. In this last framework we
consider a linear parametric generalization of WRMC, and we propose an
estimator of the explicit optimal parameter using the proposals.
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