## early rejection MCMC

In a (relatively) recent Bayesian Analysis paper on efficient MCMC algorithms for climate models, Antti Solonen, Pirkka Ollinaho, Marko Laine, Heikki Haario, Johanna Tamminen and Heikki Järvinen propose an early rejection scheme to speed up Metropolis-Hastings algorithms. The idea is to consider a posterior distribution (proportional to)

$\pi(\theta|y)= \prod_{k=1}^nL_i(\theta|y)$

such that all terms in the product are less than one and to compare the uniform u in the acceptance step of the Metropolis-Hastings algorithm to

$L_1(\theta'|y)/\pi(\theta|y),$

then, if u is smaller than the ratio, to

$L_1(\theta'|y)L_2(\theta'|y)/\pi(\theta|y),$

and so on, until the new value has been rejected or all terms have been evaluated. The scheme obviously stops earlier than the regular Metropolis-Hastings algorithm, at no significant extra cost when the product above does not factor through a sufficient statistic. Solonen et al.  suggest ordering the terms so that the computationally simpler ones are computed first. The upper bound assumption requires and is equivalent to finding the maximum on each term of the product, though, which may be costly in its own for non-standard distributions. With my students Marco Banterle and Clara Grazian, we actually came upon this paper when preparing our delayed acceptance paper as (a) it belongs to the same category of accelerated MCMC methods (delayed acceptance and early rejection are somehow synonymous!) and (b) it mentions the early prefetching papers of Brockwell (2005) and Strid (2009).

“The acceptance probability in ABC is commonly very low, and many proposals are rejected, and ER can potentially help to detect the rejections sooner.”

In the conclusion, Solonen et al. point out a possible link with ABC but, apart from the general idea of rejecting earlier by looking at a subsample or at a proxy simulation of a summary statistics, which is also the idea at the core of Dennis Prangle’s lazy ABC, there is no obvious impact on a likelihood-free method like ABC.

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