Archive for the Books Category

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.

in the time of cholera

Posted in Books, Kids, pictures, Travel with tags , , , , , , , , , on April 6, 2014 by xi'an

Bill Fitzgerald (1948-2014)

Posted in Books, Statistics, University life with tags , , , , on April 4, 2014 by xi'an

 

Just heard a very sad item of news: our colleague and friend Bill Fitzgerald, Head of Research in the Signal Processing Laboratory in the Department of Engineering at the University of Cambridge, Fellow of Christ’s College, co-founder and Chairman of Featurespace, and fanatic guitar player, passed away yesterday. He wrote one of the very first books on MCMC with Joseph Ó Ruanaidh, Numerical Bayesian Methods Applied to Signal Processing, in 1996. On a more personal level, he invited me to Cambridge for my first visit there  in 1998 and he thus was influential in introducing me to my friends Christophe Andrieu and Arnaud Doucet. Farewell, Bill!, and may the blessing of the rain be on you…

Le Monde puzzle [#860]

Posted in Books, Kids, R with tags , , , , on April 4, 2014 by xi'an

A Le Monde mathematical puzzle that connects to my awalé post of last year:

For N≤18, N balls are placed in N consecutive holes. Two players, Alice and Bob, consecutively take two balls at a time provided those balls are in contiguous holes. The loser is left with orphaned balls. What is the values of N such that Bob can win, no matter what is Alice’s strategy?

I solved this puzzle by the following R code that works recursively on N by eliminating all possible adjacent pairs of balls and checking whether or not there is a winning strategy for the other player.

topA=function(awale){
# return 1 if current player can win, 0 otherwise

  best=0
  if (max(awale[-1]*awale[-N])==1){
  #there are adjacent balls remaining

   for (i in (1:(N-1))[awale[1:(N-1)]==1]){

    if (awale[i+1]==1){
      bwale=awale
      bwale[c(i,i+1)]=0
      best=max(best,1-topA(bwale))
      }
  }}
  return(best)
 }

for (N in 2:18) print(topA(rep(1,N)))

which returns the solution

[1] 1
[1] 1
[1] 1
[1] 0
[1] 1
[1] 1
[1] 1
[1] 0
[1] 1
[1] 1
[1] 1
[1] 1
[1] 1
[1] 0
[1] 1
[1] 1
[1] 1
<pre>

(brute-force) answering the question that N=5,9,15 are the values where Alice has no winning strategy if Bob plays in an optimal manner. (The case N=5 is obvious as there always remains two adjacent 1′s once Alice removed any adjacent pair. The case N=9 can also be shown to be a lost cause by enumeration of Alice’s options.)

[more] parallel MCMC

Posted in Books, Mountains with tags , , , , , , , , , , on April 3, 2014 by xi'an

Scott Schmidler and his Ph.D. student Douglas VanDerwerken have arXived a paper on parallel MCMC the very day I left for Chamonix, prior to MCMSki IV, so it is no wonder I missed it at the time. This work is somewhat in the spirit of the parallel papers Scott et al.’s consensus Bayes,  Neiswanger et al.’s embarrassingly parallel MCMC, Wang and Dunson’s Weierstrassed MCMC (and even White et al.’s parallel ABC), namely that the computation of the likelihood can be broken into batches and MCMC run over those batches independently. In their short survey of previous works on parallelization, VanDerwerken and Schmidler overlooked our neat (!) JCGS Rao-Blackwellisation with Pierre Jacob and Murray Smith, maybe because it sounds more like post-processing than genuine parallelization (in that it does not speed up the convergence of the chain but rather improves the Monte Carlo usages one can make of this chain), maybe because they did not know of it.

“This approach has two shortcomings: first, it requires a number of independent simulations, and thus processors, equal to the size of the partition; this may grow exponentially in dim(Θ). Second, the rejection often needed for the restriction doesn’t permit easy evaluation of transition kernel densities, required below. In addition, estimating the relative weights wi with which they should be combined requires care.” (p.3)

The idea of the authors is to replace an exploration of the whole space operated via a single Markov chain (or by parallel chains acting independently which all have to “converge”) with parallel and independent explorations of parts of the space by separate Markov chains. “Small is beautiful”: it takes a shorter while to explore each set of the partition, hence to converge, and, more importantly, each chain can work in parallel to the others. More specifically, given a partition of the space, into sets Ai with posterior weights wi, parallel chains are associated with targets equal to the original target restricted to those Ai‘s. This is therefore an MCMC version of partitioned sampling. With regard to the shortcomings listed in the quote above, the authors consider that there does not need to be a bijection between the partition sets and the chains, in that a chain can move across partitions and thus contribute to several integral evaluations simultaneously. I am a bit worried about this argument since it amounts to getting a random number of simulations within each partition set Ai. In my (maybe biased) perception of partitioned sampling, this sounds somewhat counter-productive, as it increases the variance of the overall estimator. (Of course, not restricting a chain to a given partition set Ai has the incentive of avoiding a possibly massive amount of rejection steps. It is however unclear (a) whether or not it impacts ergodicity (it all depends on the way the chain is constructed, i.e. against which target(s)…) as it could lead to an over-representation of some boundaries and (b) whether or not it improves the overall convergence properties of the chain(s).)

“The approach presented here represents a solution to this problem which can completely remove the waiting times for crossing between modes, leaving only the relatively short within-mode equilibration times.” (p.4)

A more delicate issue with the partitioned MCMC approach (in my opinion!) stands with the partitioning. Indeed, in a complex and high-dimension model, the construction of the appropriate partition is a challenge in itself as we often have no prior idea where the modal areas are. Waiting for a correct exploration of the modes is indeed faster than waiting for crossing between modes, provided all modes are represented and the chain for each partition set Ai has enough energy to explore this set. It actually sounds (slightly?) unlikely that a target with huge gaps between modes will see a considerable improvement from the partioned version when the partition sets Ai are selected on the go, because some of the boundaries between the partition sets may be hard to reach with a off-the-shelf proposal. (Obviously, the second part of the method on the adaptive construction of partitions is yet in the writing and I am looking forward its aXival!)

Furthermore, as noted by Pierre Jacob (of Statisfaction fame!), the adaptive construction of the partition has a lot in common with Wang-Landau schemes. Which goal is to produce a flat histogram proposal from the current exploration of the state space. Connections with Atchadé’s and Liu’s (2010, Statistical Sinica) extension of the original Wang-Landau algorithm could have been spelled out. Esp. as the Voronoï tessellation construct seems quite innovative in this respect.

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