## MCMC with control variates

**I**n 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.

February 17, 2012 at 11:46 pm

thanks a lot.. you really do a great job in keeping us (young researchers) updated with what is going on.. it’s like having a permanent office hour for a class dealing with the most updated topics.

February 17, 2012 at 7:58 pm

Thanks, Dani, sorry if I gave the impression of disliking the paper. I actually think it is a great idea to have vanilla control variates in all settings where KG can be computed. Your suggestion is interesting in that it is sufficient to produce an unbiased estimator of KG to keep the control variate a control variate, I think, so this could lead to an extension in the spirit of Andrieu & Roberts (AoS, 2009).

February 17, 2012 at 4:23 pm

Xian, I think your comment would be legitimate even if you had been the the co-editor of that paper.

I personally see your point but on the other hand I think the Greek paper presents a clear innovation and is certainly going in the right direction. More needs to follow, this is just a beginning.

In the more general problem, do you think it could be possible to “approximate” the KG operator? (Perhaps naively) I think that an approximate control variate might be better than not having one at all.

(I don’t know the authors)