important Markov chains

With Charly Andral (PhD, Paris Dauphine), Randal Douc, and Hugo Marival (PhD, Telecom SudParis), we just arXived a paper on importance Markov chains that merges importance sampling and MCMC. An idea already mentioned in Hastings (1970) and even earlier in Fodsick (1963), and later exploited in Liu et al.  (2003) for instance. And somewhat dual of the vanilla Rao-Backwellisation paper Randal and I wrote a (long!) while ago. Given a target π with a dominating measure π⁰≥Mπ, using a Markov kernel to simulate from this dominating measure and subsampling by the importance weight ρ does produce a new Markov chain with the desired target measure as invariant distribution. However, the domination assumption is rather unrealistic and a generic approach can be implemented without it, by defining an extended Markov chain, with the addition of the number N of replicas as the supplementary term… And a transition kernel R(n|x) on N with expectation ρ, which is a minimal(ist) assumption for the validation of the algorithm.. While this initially defines a semi-Markov chain, an extended Markov representation is also feasible, by decreasing N one by one until reaching zero, and this is most helpful in deriving convergence properties for the resulting chain, including a CLT.  While the choice of the kernel R is free, the optimal choice is associated with residual sampling, where only the fractional part of ρ is estimated by a Bernoulli simulation.

informed proposals for local MCMC in discrete spaces

Last year Giacomo Zanella published a paper entitled informed proposals for local MCMC in discrete spaces in JASA. Which I had missed somehow and only discovered through another paper, and which we recently discussed at Paris-Dauphine with graduate students, marooned by COVID-19 . Probability targets in discrete spaces are intrinsically hard[er] to simulate in my opinion if only because there is no natural distance, hence no natural neighbourhood. A random walk proposal like the reference kernel in the paper is not directly calibrated. Without demarginalisation there is neither a clear version of calculus for implementing MALA or HMC. What indeed is HMC on a discrete space? If this requires “embedding the binary space in a continuous space”, it does not sound very enticing if the construct is context dependent.

“This would allow for more moves to be accepted and longer moves to be performed, thus improving the algorithm’s efficiency.”

A interesting aspect of the paper is that for near atomic transition kernels K, informally for small σ’s, the proposal switch to Q finds target x normalising constant as new stationary and close to the actual target. Which incidentally reminded me of our vanilla Rao-Blackwellisation with Randal Douc. This however begets the worry that it may prove unwieldy in continuous cases, as except for Gaussian kernels, the  proposal switch to Q may prove intractable and requires further MCMC steps, in a form of infinite regress. Plus a musing that, were the original kernel K to be replaced with the new Q, another informed proposal transform could be applied to Q. Further infinite regress…

“[The optimality of the Metropolis-Hastings choice of acceptance probability] does not translate to the context of balancing functions.”

The paper indeed exhibits a setting that is rehabilitating Barker’ (1965) version of the acceptance probability, but I never  was very much convinced there was a significant difference in using one or the other. During our virtual (?) discussion, we also wondered at the adaptive abilities of the approach, e.g., selecting among a finite family of g’s (according to which criterion) or parameterising g towards an optimal choice of its parameter. And at the capacity for Rao-Blackwellisation since the proposal have to consider the entire set of neighbours prior to moving to a likely one.

thinning a Markov chain, statistically

Art Owen has arXived a new version of his thinning MCMC paper, where he studies how thinning or subsampling can improve computing time in MCMC chains. I remember quite well the message set by Mark Berliner and Steve MacEachern in an early 1990’s paper that subsampling was always increasing the variance of the resulting estimators. We actually have this result in our Monte Carlo Statistical Methods book. Now, there are other perspectives on this, as for instance cases when thinning can be hard-wired by simulating directly a k-step move, delaying rejection or acceptance, prefetching, or simulating directly the accepted values as in our vanilla Rao-Blackwellisation approach. Here, Art considers the case when there is a cost θ of computing a transform of the simulation [when the transition cost a unit] and when those transforms are positively correlated with correlation ρ. Somewhat unsurprisingly, when θ is large enough, thinning becomes worth implementing. But requires extra computations in evaluating the correlation ρ and the cost θ, which is rarely comparable with the cost of computing the likelihood itself, a requirement for the Metropolis-Hastings or Hamiltonian Monte Carlo step(s).  Subsampling while keeping the right target (which is a hard constraint!) should thus have a much more effective impact on computing budgets.

MCMskv #4 [house with a vision]

Last day at MCMskv! Not yet exhausted by this exciting conference, but this was the toughest day with one more session and a tutorial by Art Own on quasi Monte-Carlo. (Not even mentioning the night activities that I skipped. Or the ski break that I did not even consider.) Krys Latunszynski started with a plenary on exact methods for discretised diffusions, with a foray in Bernoulli factory problems. Then a neat session on adaptive MCMC methods that contained a talk by Chris Sherlock on delayed acceptance, where the approximation to the target was built by knn trees. (The adaptation was through the construction of the tree by including additional evaluations of the target density. Another paper sitting in my to-read list for too a long while: the exploitation of the observed values of π towards improving an MCMC sampler has always be “obvious” to me even though I could not see any practical way of doing so. )

It was wonderful that Art Owen accepted to deliver a tutorial at MCMskv on quasi-random Monte Carlo. Great tutorial, with a neat coverage of the issues most related to Monte Carlo integration. Since quasi-random sequences have trouble with accept/reject methods, a not-even-half-baked idea that came to me during Art’s tutorial was that the increased computing power granted by qMC could lead to a generic integration of the Metropolis-Hastings step in a Rao-Blackwellised manner. Art mentioned he was hoping that in a near future one could switch between pseudo- and quasi-random in an almost automated manner when running standard platforms like R. This would indeed be great, especially since quasi-random sequences seem to be available at the same cost as their pseudo-random counterpart. During the following qMC session, Art discussed the construction of optimal sequences on sets other than hypercubes (with the surprising feature that projecting optimal sequences from the hypercube does not work). Mathieu Gerber presented the quasi-random simulated annealing algorithm he developed with Luke Bornn that I briefly discussed a while ago. Or thought I did as I cannot trace a post on that paper! While the fact that annealing also works with quasi-random sequences is not astounding, the gain over random sequences shown on two examples is clear. The session also had a talk by Lester Mckey who relies Stein’s discrepancy to measure the value of an approximation to the true target. This was quite novel, with a surprising connection to Chris Oates’ talk and the use of score-based control variates, if used in a dual approach.

Another great session was the noisy MCMC one organised by Paul Jenkins (Warwick), with again a coherent presentation of views on the quality or lack thereof of noisy (or inexact) versions, with an update from Richard Everitt on inexact MCMC, Felipe Medina Aguayo (Warwick) on sufficient conditions for noisy versions to converge (and counterexamples), Jere Koskela (Warwick) on a pseudo-likelihood approach to the highly complex Kingman’s coalescent model in population genetics (of ABC fame!), and Rémi Bardenet on the tall data approximations techniques discussed in a recent post. Having seen or read most of those results previously did not diminish the appeal of the session.

locally weighted MCMC

Last week, on arXiv, Espen Bernton, Shihao Yang, Yang Chen, Neil Shephard, and Jun Liu (all from Harvard) proposed a weighting scheme to associated MCMC simulations, in connection with the parallel MCMC of Ben Calderhead discussed earlier on the ‘Og. The weight attached to each proposal is either the acceptance probability itself (with the rejection probability being attached to the current value of the MCMC chain) or a renormalised version of the joint target x proposal, either forward or backward. Both solutions are unbiased in that they have the same expectation as the original MCMC average, being some sort of conditional expectation. The proof of domination in the paper builds upon Calderhead’s formalism.

This work reminded me of several reweighting proposals we made over the years, from the global Rao-Blackwellisation strategy with George Casella, to the vanilla Rao-Blackwellisation solution we wrote with Randal Douc a few years ago, both of whom also are demonstrably improving upon the standard MCMC average. By similarly recycling proposed but rejected values. Or by diminishing the variability due to the uniform draw. The slightly parallel nature of the approach also connects with our parallel MCM version with Pierre Jacob (now Harvard as well!) and Murray Smith (who now leaves in Melbourne, hence the otherwise unrelated picture).