Matti Vihola from the University of Jyväskylä posted a paper on arXiv yesterday on a convergence result for an adaptive scheme related with the basic random walk Metropolis-Hastings algorithm. The scale$\theta$used in the random walk is adapted using the Robbins-Monro stochastic approximation schedule
$\log \theta_{t+1} = \log\theta_t + Ct^{-\gamma}(\alpha_t-\alpha^\star)$
where$\alpha^\star$is the gold standard for acceptance, and$\alpha_t$is the empirical acceptance rate. (This was one of the examples in our 2001 paper with Christophe Andrieu, Controlled MCMC for Optimal Sampling, that never got published but can nonetheless boasts about its 43 citations!) The constraints imposed upon the target density$\pi$and on the integrand$f$are fairly harsh, including$f$being bounded (those constraints seem to be more restrictive than in Roberts and Rosenthal, 2007, J.A.P., although of course I did not check the correspondance) and there is a surprising restriction on$\alpha^\star$, namely$\alpha^\star<1/2$. I do not have any intuitive explanation for this hard boundary on the acceptance rate: staying away from 0 and from 1 makes sense, obviously, but 1/2? Looking at the reason, this seems to be related to a lower bound on the average acceptance rate found in the proof of Proposition 12, which is itself related with the convergence of the Robbins-Monro sequence.
Ps-When checking for the role of$\alpha$, I came across the possibility to search for$\alpha$in a pdf file with Acrobat Reader by simply typing$\alpha$in the Find box. Neat!