2006/11/30 by Jean‐François Delmas, Benjamin Jourdain, Delmas, Jean-François +1
Computer Science · Mathematics · Physics and Astronomy · #60F05 #60J10 #60J22 #65C40 #82B80 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic Gradient Optimization Techniques #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.math/0611949
openalex publication_date 2006/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 π difficult to simulate. They consist in constructing by an appropriate acceptation/rejection procedure a Markov chain (Xk,k≥ 0) with transition matrix P such that π is reversible with respect to P and in estimating by the empirical mean In(f)=\invn∑k=1n f(Xk). 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.