Using all Metropolis-Hastings proposals to estimate an integral Suppose we run the Metropolis-Hastings with target distribution $\mu$ to compute the integral $\int f\:{\rm d}\mu$. Usually, we use the estimator $$A_n:=\frac1n\sum_{i=0}^{n-1}f(X_i).$$ However, instead of adding up the $f(X_i)$, I've seen the following modification: Let $(Y_n)_n$ denote the sequence of proposals. By definition, $Y_i$ is accepted and $X_i$ is set to $Y_i$ with probability $\alpha(X_{i-1},X_i)$. Instead of adding $f(X_i)$ to the sum, we could add $(1-\alpha(X_{i-1},Y_i))f(X_{i-1})+\alpha(X_{i-1},Y_i)f(Y_i)$.
I couldn't find any lecture book or paper considering this modification or an estimator modified in a similar way (except this paper). Does anybody have a reference at hand?
 A: Recycling proposed values in a Metropolis-Hastings algorithm goes under the name of Rao-Blackwellisation. For instance, we made such a proposal in 


*

*Casella and Robert (1996) Rao-Blackwellisation of sampling schemes. 

*Douc and Robert (2013) A vanilla Rao-Blackwellisation of Metropolis-Hastings algorithms
Note that the weighting you propose in the question:
$$[1-\alpha(x_i,y_{i+1})]h(x_i)+\alpha(x_i,y_{i+1})h(y_i)$$
is not optimal in that it is not guaranteed to bring the variance down. Even the one I produce [above] in one of my MCMC course slides does not always reduce the variance because of the correlation between the different terms. (Only the data augmentation case leads to a sure reduction in the variance, cf this fantastic paper by Liu, Wong and Long, 1994. And vanilla Rao-Blackwellisation.)
A: This looks similar to "local" importance sampling. In the literature, constructing estimators of this sort seem to termed as "waste-recyling", and quick search yields a few papers:


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*This thesis by Murray

*PNAS paper by Freknel here

*Does waste-recyling help by Delmas and Jourdain here

*Their use in parallel MCMC here
Further search using the keyword "waste-recyling" and look within the above references might give you some more papers.
