## Archive for quasi-Monte Carlo methods

## QMC at CIRM

Posted in Mountains, pictures, Statistics, Travel, University life with tags calanques, CIRM, determinism, Diophantine problems, Jean Morlet Chair, Luminy campus, Marseiile, qMC, quasi-Monte Carlo methods, randomness, research school, Université Aix Marseille on October 21, 2020 by xi'an## MCqMC 2020 live and free and online

Posted in pictures, R, Statistics, Travel, University life with tags COVID-19, England, Markov chain Monte Carlo algorithm, MCQMC 2020, Monte Carlo methods, Monte Carlo Statistical Methods, Oxford, quasi-Monte Carlo methods, remote conference, tutorial, University of Oxford, virtual reality, webinar, youtube on July 27, 2020 by xi'an**T**he MCqMC 20202 conference that was supposed to take place in Oxford next 9-14 August has been turned into an on-line free conference since travelling remains a challenge for most of us. Tutorials and plenaries will be live with questions on Zoom, with live-streaming and recorded copies on YouTube. They will probably be during 14:00-17:00 UK time (GMT+1), 15:00-18:00 CET (GMT+2), and 9:00-12:00 ET. (Which will prove a wee bit of a challenge for West Coast and most of Asia and Australasia researchers, which is why our One World IMS-Bernoulli conference we asked plenary speakers to duplicate their talks.) All other talks will be pre-recorded by contributors and uploaded to a website, with an online Q&A discussion section for each. As a reminder here are the tutorials and plenaries:

Invited plenary speakers:

Aguêmon Yves Atchadé (Boston University)

Jing Dong (Columbia University)

Pierre L’Écuyer (Université de Montréal)

Mark Jerrum (Queen Mary University London)

Peter Kritzer (RICAM Linz)

Thomas Muller (NVIDIA)

David Pfau (Google DeepMind)

Claudia Schillings (University of Mannheim)

Mario Ullrich (JKU Linz)

Tutorials:

Fred Hickernell (IIT) — Software for Quasi-Monte Carlo Methods

Aretha Teckentrup (Edinburgh) — Markov chain Monte Carlo methods

## Markov chain quasi-Monte Carlo

Posted in Statistics with tags congruential generators, Mersenne generator, pseudo-random generator, quasi-Monte Carlo methods, Tausworthe generator on April 29, 2020 by xi'an

“It is known that Tausworthe generators can be viewed as polynomial Korobov lattice point sets with a denominator polynomial p(x) and a numerator polynomial q(x) over IF_{2}“

**A** recently arXived paper by Shin Harase, “A table of short-period Tausworthe generators for Markov chain quasi-Monte Carlo”, discusses the use of [quasi-Monte Carlo] Tausworthe generators rather than iid uniform sampling. As shown by Owen and Tribble, it is indeed legit to replace a sequence of iid (pseudo-random) uniforms with its quasi-Monte Carlo (qMC) version if the sequence keeps a sufficient degree of uniformity. The current paper optimises the parameters of the Tausworthe generators in terms of the *t-*value of the generator, an indicator of the uniform occupancy of the qMC sequence.

For a range of values of *m*, if *2 ^{m}-1* is the period of the pseudo-random generator, the author obtains the optimal weights in the Tausworthe generator, which is a linear feedback shift register generator over {0,1}, ie shifting all the bits of the current uniform realisation by linear combination modulo 2. The comparison with other qMC and MC is provided on a Gibbs sampler for a bidimensional Gaussian target, which presents the advantage of requiring exactly one uniform per simulation and the disadvantage of … requiring exactly one uniform per simulation! Since this is harder to envision for simulation methods requiring a random number of uniforms.

Regarding the complexity of the approach, I do not see any gap between using these Tausworthe generators and something like the Mersenne generator. I just wonder at the choice of *m*, that is, whether or not it makes sense to pick any value lower than 2³² for the period.

## MCqMC2020 key dates

Posted in pictures, Statistics, Travel, University life with tags Britain, Eagle and Child, MCMC2020, Monte Carlo methods, Oxford, Oxford Mathematics, quasi-Monte Carlo methods, Saint Giles cemetery, scientific computing, University of Oxford on January 23, 2020 by xi'an**A** reminder of the key dates for the incoming MCqMC2020 conference this summer in Oxford:

Feb 28, Special sessions/minisymposia submission

Mar 13, Contributed abstracts submission

Mar 27, Acceptance notification

Mar 27, Registration starts

May 8, End of early bird registration

June 12, Speaker registration deadline

Aug 9-14 Conference

and of the list of plenary speakers

Yves Atchadé (Boston University)

Jing Dong (Columbia University)

Pierre L’Ecuyer (Université de Montreal)

Mark Jerrum (Queen Mary University London)

Gerhard Larcher (JKU Linz)

Thomas Muller (NVIDIA)

David Pfau (Google DeepMind)

Claudia Schillings (University of Mannheim)

Mario Ullrich (JKU Linz)

## ABC by QMC

Posted in Books, Kids, Statistics, University life with tags ABC, ABC-PMC, ABC-SMC, CREST, JCGS, PhD thesis, population genetics, population Monte Carlo, qMC, quasi-Monte Carlo methods, variance reduction on November 5, 2018 by xi'an**A** paper by Alexander Buchholz (CREST) and Nicolas Chopin (CREST) on quasi-Monte Carlo methods for ABC is going to appear in the *Journal of Computational and Graphical Statistics*. I had missed the opportunity when it was posted on arXiv and only became aware of the paper’s contents when I reviewed Alexander’s thesis for the doctoral school. The fact that the parameters are simulated (in ABC) from a prior that is quite generally a standard distribution while the pseudo-observations are simulated from a complex distribution (associated with the intractability of the likelihood function) means that the use of quasi-Monte Carlo sequences is in general only possible for the first part.

The ABC context studied there is close to the original version of ABC rejection scheme [as opposed to SMC and importance versions], the main difference standing with the use of M pseudo-observations instead of one (of the same size as the initial data). This repeated version has been discussed and abandoned in a strict Monte Carlo framework in favor of M=1 as it increases the overall variance, but the paper uses this version to show that the multiplication of pseudo-observations in a quasi-Monte Carlo framework does not increase the variance of the estimator. (Since the variance apparently remains constant when taking into account the generation time of the pseudo-data, we can however dispute the interest of this multiplication, except to produce a constant variance estimator, for some targets, or to be used for convergence assessment.) L The article also covers the bias correction solution of Lee and Latuszyǹski (2014).

Due to the simultaneous presence of pseudo-random and quasi-random sequences in the approximations, the authors use the notion of mixed sequences, for which they extend a one-dimension central limit theorem. The paper focus on the estimation of Z(ε), the normalization constant of the ABC density, ie the predictive probability of accepting a simulation which can be estimated at a speed of O(N⁻¹) where N is the number of QMC simulations, is a wee bit puzzling as I cannot figure the relevance of this constant (function of ε), especially since the result does not seem to generalize directly to other ABC estimators.

A second half of the paper considers a sequential version of ABC, as in ABC-SMC and ABC-PMC, where the proposal distribution is there based on a Normal mixture with a *small* number of components, estimated from the (particle) sample of the previous iteration. Even though efficient techniques for estimating this mixture are available, this innovative step requires a calculation time that should be taken into account in the comparisons. The construction of a decreasing sequence of tolerances ε seems also pushed beyond and below what a sequential approach like that of Del Moral, Doucet and Jasra (2012) would produce, it seems with the justification to always prefer the lower tolerances. This is not necessarily the case, as recent articles by Li and Fearnhead (2018a, 2018b) and ours have shown (Frazier et al., 2018). Overall, since ABC methods are large consumers of simulation, it is interesting to see how the contribution of QMC sequences results in the reduction of variance and to hope to see appropriate packages added for standard distributions. However, since the most consuming part of the algorithm is due to the simulation of the pseudo-data, in most cases, it would seem that the most relevant focus should be on QMC add-ons on this part, which may be feasible for models with a huge number of standard auxiliary variables as for instance in population evolution.