Advances in scalable Bayesian computation [day #4]

Final day of our workshop Advances in Scalable Bayesian Computation already, since tomorrow morning is an open research time ½ day! Another “perfect day in paradise”, with the Banff Centre campus covered by a fine snow blanket, still falling…, and making work in an office of BIRS a dream-like moment.

Still looking for a daily theme, parallelisation could be the right candidate, even though other talks this week went into parallelisation issues, incl. Steve’s talk yesterday. Indeed, Anthony Lee gave a talk this morning on interactive sequential Monte Carlo, where he motivated the setting by a formal parallel structure. Then, Darren Wilkinson surveyed the parallelisation issues in Monte Carlo, MCMC, SMC and ABC settings, before arguing in favour of a functional language called Scala. (Neat entries to those topics can be found on Darren’s blog.) And in the afternoon session, Sylvia Frühwirth-Schnatter exposed her approach to the (embarrassingly) parallel problem, in the spirit of Steve’s , David Dunson’s and Scott’s (a paper posted on the day I arrived in Chamonix and hence I missed!). There was plenty to learn from that talk (do not miss the Yin-Yang moment at 25 mn!), but it also helped me to break a difficulty I had with the consensus Bayes representation for two weeks (more on that later!). And, even though Marc Suchard mostly talked about flu and trees in a very pleasant and broad talk, he also had a slide on parallelisation to fit the theme! Although unrelated with parallelism,  Nicolas Chopin’s talk was on sequential quasi-Monte Carlo algorithms: while I had heard previous versions of this talk in Chamonix and BigMC, I found it full of exciting stuff. And it clearly got the room truly puzzled by this possibility, in a positive way! Similarly, Alex Lenkoski spoke about extreme rain events in Norway with no trace of parallelism, but the general idea behind the examples was to question the notion of the calibrated Bayesian (with possible connections with the cut models).

This has been a wonderful week and I am sure the participants got as much as I did from the talks and the informal exchanges. Thanks to BIRS for the sponsorship and the superb organisation of the week (and to the Banff Centre for providing such a paradisical environment). I feel very privileged to have benefited from this support, even though I deadly hope to be back in Banff within a few years.

Winter workshop, Gainesville (day 2)

On day #2, besides my talk on “empirical Bayes” (ABCel) computation (mostly recycled from Varanasi, photos included), Christophe Andrieu gave a talk on exact approximations, using unbiased estimators of the likelihood and characterising estimators garanteeing geometric convergence (bounded weights, essentially, which is a condition popping out again and again in the Monte Carlo literature). Then Art Owen (father of empirical likelihood among other things!) spoke about QMC for MCMC, a topic that always intringued me.

Indeed, while I see the point of using QMC for specific integration problems, I am more uncertain about its relevance for statistics as a simulation device. Having points uniformly distributed over the unit hypercube in a much more efficient way than a random sample is not helping much when only a tiny region of the unit hypercube, namely the one where the likelihood concentrates, matters. (In other words, we are rarely interested in the uniform distribution over the unit hypercube: we instead want to simulate from a highly irregular and definitely concentrated distribution.) I have the same reservation about the applicability of stratified sampling: the strata have to be constructed in relation with the target distribution. The method Art advocates using a CUD (completely uniformly distributed) sequence as the underlying (deterministic) pseudo-unifom sequence. Highly interesting and I want to read the paper in greater details, but the fact that most simulation steps use a random number of uniforms seems detrimental to the performances of the method in general.

After a lunch break at a terrific BBQ place, with a stop at Lake Alice to watch the alligator(s) I had missed during my morning run, I was able this time to attend till the end Xiao-Li Meng’s talk, where he presented new improvements on bridge sampling based on location-scale (or warping) transforms of the original two-samples to make them share mean and variance. Hani Doss concluded the meeting with a talk on the computation of Bayes factors when using (non-parametric) Dirichlet mixture priors, whose resolution does not require simulations for each value of the scale parameter of the Dirichlet prior, thanks to a Radon-Nykodim derivative representation. (Which nicely connected with Art’s talk in that the latter mentioned therein that most simulation methods are actually based on Riemann integration rather than Lebesgue integration. Hani’s representation is not, with nested sampling being another example.)

We ended up the day with a(nother) barbecue outside, under the stars, in the peace and quiet of a local wood, with wine and laughs, just like George would have concluded the workshop. This was a fitting ending to a meeting dedicated to his memory…

A survey of the [60′s] Monte Carlo methods [2]

The 24 questions asked by John Halton in the conclusion of his 1970 survey are

1. Can we obtain a theory of convergence for random variables taking values in Fréchet spaces?
2. Can the study of Monte Carlo estimates in separable Fréchet spaces give a theory of global approximation?
3. When sampling functions, what constitutes a representative sample of function values?
4. Can one apply Monte Carlo to pattern recognition?
5. Relate Monte Carlo theory to the theory of random equations.
6. What can be said about quasi-Monte Carlo estimates for finite-dimensional and infinite-dimensional integrals?
7. Obtain expression, asymptotic forms or upper bounds for L² and L^∞ discrepancies of quasirandom sequences.
8. How should one improve quasirandom sequences?
9. How to interpret the results of statistical tests applied to pseudo- or quasirandom sequences?
10. Can we develop a meaningful statistical theory of quasi-Monte Carlo estimates?
11. Can existing Monte Carlo techniques be improved and applied to new classes of problems?
12. Can the design of Monte Carlo estimators be made more systematic?
13. How can the idea of sequential Monte Carlo be extended?
14. Can sampling with signed probabilities be made practical?
15. What is the best allocation effort in obtaining zeroth- and first-level estimators in algebraic problems?
16. Examine the Monte Carlo analogues of the various matrix iterative schemes.
17. Develop the schemes of grid refinement in continuous problems.
18. Develop new Monte Carlo eigenvectors and eigenvalue techniques.
19. Develop fast, reliable true canonical random generators.
20. How is the output of a true random generator to be tested?
21. Develop fast, efficient methods for generating arbitrary random generators.
22. Can we really have useful general purpose pseudorandom sequences.
23. What is the effect of the discreteness of digital computers on Monte Carlo calculations?
24. Is there a way to estimate the accuracy of Monte Carlo estimates?