The Metropolis Hastings algorithm and its multi-proposal extensions are aimed
at the computation of the expectation of a function $f$ under a
probability measure $\pi$ difficult to simulate. They consist in constructing
by an appropriate acceptation/rejection procedure a Markov chain $(X_k,k\geq
0)$ with transition matrix $P$ such that $\pi$ is reversible with respect to
$P$ and in estimating by the empirical mean
I_n(f)=\inv{n}\sum_{k=1}^n f(X_k). The waste-recycling Monte Carlo (WR)
algorithm introduced by physicists is a modification of the Metropolis-Hastings
algorithm, which makes use of all the proposals in the empirical mean, whereas
the standard Metropolis-Hastings algorithm only uses the accepted proposals. In
this paper, we extend the WR algorithm into a general control variate technique
and exhibit the optimal choice of the control variate in terms of asymptotic
variance. We also give an example which shows that in contradiction to the
intuition of physicists, the WR algorithm can have an asymptotic variance
larger than the one of the Metropolis-Hastings algorithm. However, in the
particular case of the Metropolis-Hastings algorithm called Boltzmann
algorithm, we prove that the WR algorithm is asymptotically better than the
Metropolis-Hastings algorithm
