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early rejection MCMC

Posted in Books, Statistics, University life with tags , , , , , , , , on June 16, 2014 by xi'an

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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delayed acceptance with prefetching

Posted in Books, Kids, Statistics, Travel, University life with tags , , , , , , , on June 12, 2014 by xi'an

In a somewhat desperate rush (started upon my return from Iceland and terminated on my return from Edinburgh), Marco Banterle, Clara Grazian and I managed to complete and submit our paper by last Friday evening… It is now arXived as well. The full title of the paper is Accelerating Metropolis-Hastings algorithms: Delayed acceptance with prefetching and the idea behind the generic acceleration is (a) to divide the acceptance step into parts, towards a major reduction in computing time that outranks the corresponding reduction in acceptance probability and (b) to exploit this division to build a dynamic prefetching algorithm. The division is to break the prior x likelihood target into a product such that some terms are much cheaper than others. Or equivalently to represent the acceptance-rejection ratio in the Metropolis-Hastings algorithm as

\dfrac{\pi(\theta)\,q(\theta,\eta)}{\pi(\eta)q(\eta,\theta)} = \prod_{k=1}^d \rho_k(\eta,\theta)

again with significant differences in the computing cost of those terms. Indeed, this division can be exploited by checking for each term sequentially, in the sense that the overall acceptance probability

\prod_{k=1}^d \min\left\{\rho_k(\eta,\theta),1\right\}

is associated with the right (posterior) target! This lemma can be directly checked via the detailed balance condition, but it is also a consequence of a 2005 paper by Andrès Christen and Colin Fox on using approximate transition densities (with the same idea of gaining time: in case of an early rejection, the exact target needs not be computed). While the purpose of the recent [commented] paper by Doucet et al. is fundamentally orthogonal to ours, a special case of this decomposition of the acceptance step in the Metropolis–Hastings algorithm can be found therein. The division of the likelihood into parts also allows for a precomputation of the target solely based on a subsample, hence gaining time and allowing for a natural prefetching version, following recent developments in this direction. (Discussed on the ‘Og.) We study the novel method within two realistic environments, the fi rst one made of logistic regression targets using benchmarks found in the earlier prefetching literature and a second one handling an original analysis of a parametric mixture model via genuine Jeffreys priors. [As I made preliminary notes along those weeks using the ‘Og as a notebook, several posts on the coming days will elaborate on the above.]

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accelerating MCMC via parallel predictive prefetching

Posted in Books, Statistics, University life with tags , , , , , , , , on April 7, 2014 by xi'an

¨The idea is to calculate multiple likelihoods ahead of time (“pre-fetching”), and only use the ones which are needed.” A. Brockwell, 2006

Yet another paper on parallel MCMC, just arXived by Elaine Angelino, Eddie Kohler, Amos Waterland, Margo Seltzer, and Ryan P. Adams. Now,  besides “prefetching” found in the title, I spotted “speculative execution”, “slapdash treatment”, “scheduling decisions” in the very first pages: this paper definitely is far from shying away from using fancy terminology! I actually found the paper rather difficult to read to the point I had to give up my first attempt during an endless university board of governors meeting yesterday. (I also think “prefetching” is awfully painful to type!)

What is “prefetching” then? It refers to a 2006 JCGS paper by Anthony Brockwell. As explained in the above quote from Brockwell, prefetching means computing the 2², 2³, … values of the likelihood that will be needed in 2, 3, … iterations. Running a regular Metropolis-Hastings algorithm then means building a decision tree back to the current iteration and drawing 2,3, … uniform to go down the tree to the appropriate branch. So in the end only one path of the tree is exploited, which does not seem particularly efficient when vanilla Rao-Blackwellisation and recycling could be implemented almost for free.

“Another intriguing possibility, suggested to the author by an anonymous referee, arises in the case where one can guess whether or not acceptance probabilities will be “high” or “low.” In this case, the tree could be made deeper down “high” probability paths and shallower in the “low” probability paths.” A. Brockwell, 2006

The current paper stems from Brockwell’s 2006 final remark, as reproduced above, by those “speculative moves” that considers the reject branch of the prefetching tree more often that not, based on some preliminary or dynamic evaluation of the acceptance rate. Using a fast but close enough approximation to the true target (and a fixed sequence of uniforms) may also produce a “single most likely path on which” prefetched simulations can be run. The basic idea is thus to run simulations and costly likelihood computations on many parallel processors along a prefetched path, path that has been prefetched for its high approximate likelihood. (With of courses cases where this speculative simulation is not helpful because we end up following another path with the genuine target.) The paper actually goes further than the basic idea to avoid spending useless time on paths that will not be chosen, by constructing sequences of approximations for the precomputations. The proposition for the sequence found therein is to subsample the original data and use a normal approximation to the difference of the log (sub-)likelihoods. Even though the authors describe the system implementation of the progressive approximation idea, it remains rather unclear (to me) how the adaptive estimation of the acceptance probability is compatible with the parallelisation idea. Because it seems (to me) that it induces a lot of communication between the cores. Also, the method is advocated mainly for burnin’ (or warmup, to follow Andrew’s terminology!), which seems to remove the need to use exact targets: if the approximation is close enough, the Markov chain will quickly reach a region of interest for the true target and from there there seems to be little speedup in implementing this nonetheless most interesting strategy.

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