## MCMC with control variates

Posted in Books, Statistics, University life with tags , , , , , , , , , , on February 17, 2012 by xi'an

In the latest issue of JRSS Series B (74(1), Jan, 2012), I just noticed that no paper is “from my time” as co-editor, i.e. that all of them have been submitted after I completed my term in Jan. 2010. Given the two year delay, this is not that surprising, but it also means I can make comments on some papers w/o reservation! A paper I had seen earlier (as a reader, not as an editor nor as a referee!) is Petros Dellaportas’ and Ioannis Kontoyiannis’ Control variates for estimation based on  reversible Markov chain Monte Carlo samplers. The idea is one of post-processing MCMC output, by stabilising the empirical average via control variates. There are two difficulties, one in finding control variates, i.e. functions $\Psi(\cdot)$ with zero expectation under the target distribution, and another one in estimating the optimal coefficient in a consistent way. The paper solves the first difficulty by using the Poisson equation, namely that G(x)-KG(x) has zero expectation under the stationary distribution associated with the Markov kernel K. Therefore, if KG can be computed in closed form, this is a generic control variate taking advantage of the MCMC algorithm. Of course, the above if is a big if: it seems difficult to find closed form solutions when using a Metropolis-Hastings algorithm for instance and the paper only contains illustrations within the conjugate prior/Gibbs sampling framework. The second difficulty is also met by Dellaportas and Kontoyiannis, who show that the asymptotic variance of the resulting central limit can be equal to zero in some cases.