Archive for quasi-Monte Carlo methods

ABC by QMC

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , , , 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.

impressions from EcoSta2017 [guest post]

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , on July 6, 2017 by xi'an

[This is a guest post on the recent EcoSta2017 (Econometrics and Statistics) conference in Hong Kong, contributed by Chris Drovandi from QUT, Brisbane.]

There were (at least) two sessions on Bayesian Computation at the recent EcoSta (Econometrics and Statistics) 2017 conference in Hong Kong. Below is my review of them. My overall impression of the conference is that there were lots of interesting talks, albeit a lot in financial time series, not my area. Even so I managed to pick up a few ideas/concepts that could be useful in my research. One criticism I had was that there were too many sessions in parallel, which made choosing quite difficult and some sessions very poorly attended. Another criticism of many participants I spoke to was that the location of the conference was relatively far from the city area.

In the first session (chaired by Robert Kohn), Minh-Ngoc Tran spoke about this paper on Bayesian estimation of high-dimensional Copula models with mixed discrete/continuous margins. Copula models with all continuous margins are relatively easy to deal with, but when the margins are discrete or mixed there are issues with computing the likelihood. The main idea of the paper is to re-write the intractable likelihood as an integral over a hypercube of ≤J dimensions (where J is the number of variables), which can then be estimated unbiasedly (with variance reduction by using randomised quasi-MC numbers). The paper develops advanced (correlated) pseudo-marginal and variational Bayes methods for inference.

In the following talk, Chris Carter spoke about different types of pseudo-marginal methods, particle marginal Metropolis-Hastings and particle Gibbs for state space models. Chris suggests that a combination of these methods into a single algorithm can further improve mixing. Continue reading

Monte Carlo with determinantal processes [reply from the authors]

Posted in Books, Statistics with tags , , , , , , , , , , , , , , on September 22, 2016 by xi'an

[Rémi Bardenet and Adrien Hardy have written a reply to my comments of today on their paper, which is more readable as a post than as comments, so here it is. I appreciate the intention, as well as the perfect editing of the reply, suited for a direct posting!]

Thanks for your comments, Xian. As a foreword, a few people we met also had the intuition that DPPs would be relevant for Monte Carlo, but no result so far was backing this claim. As it turns out, we had to work hard to prove a CLT for importance-reweighted DPPs, using some deep recent results on orthogonal polynomials. We are currently working on turning this probabilistic result into practical algorithms. For instance, efficient sampling of DPPs is indeed an important open question, to which most of your comments refer. Although this question is out of the scope of our paper, note however that our results do not depend on how you sample. Efficient sampling of DPPs, along with other natural computational questions, is actually the crux of an ANR grant we just got, so hopefully in a few years we can write a more detailed answer on this blog! We now answer some of your other points.

“one has to examine the conditions for the result to operate, from the support being within the unit hypercube,”
Any compactly supported measure would do, using dilations, for instance. Note that we don’t assume the support is the whole hypercube.

“to the existence of N orthogonal polynomials wrt the dominating measure, not discussed here”
As explained in Section 2.1.2, it is enough that the reference measure charges some open set of the hypercube, which is for instance the case if it has a density with respect to the Lebesgue measure.

“to the lack of relation between the point process and the integrand,”
Actually, our method depends heavily on the target measure μ. Unlike vanilla QMC, the repulsiveness between the quadrature nodes is tailored to the integration problem.

“changing N requires a new simulation of the entire vector unless I missed the point.”
You’re absolutely right. This is a well-known open issue in probability, see the discussion on Terence Tao’s blog.

“This requires figuring out the upper bounds on the acceptance ratios, a “problem-dependent” request that may prove impossible to implement”
We agree that in general this isn’t trivial. However, good bounds are available for all Jacobi polynomials, see Section 3.

“Even without this stumbling block, generating the N-sized sample for dimension d=N (why d=N, I wonder?)”
This is a misunderstanding: we do not say that d=N in any sense. We only say that sampling from a DPP using the algorithm of [Hough et al] requires the same number of operations as orthonormalizing N vectors of dimension N, hence the cubic cost.

1. “how does it relate to quasi-Monte Carlo?”
So far, the connection to QMC is only intuitive: both rely on well-spaced nodes, but using different mathematical tools.

2. “the marginals of the N-th order determinantal process are far from uniform (see Fig. 1), and seemingly concentrated on the boundaries”
This phenomenon is due to orthogonal polynomials. We are investigating more general constructions that give more flexibility.

3. “Is the variance of the resulting estimator (2.11) always finite?”
Yes. For instance, this follows from the inequality below (5.56) since ƒ(x)/K(x,x) is Lipschitz.

4. and 5. We are investigating concentration inequalities to answer these points.

6. “probabilistic numerics produce an epistemic assessment of uncertainty, contrary to the current proposal.”
A partial answer may be our Remark 2.12. You can interpret DPPs as putting a Gaussian process prior over ƒ and sequentially sampling from the posterior variance of the GP.

Monte Carlo with determinantal processes

Posted in Books, Statistics with tags , , , , , , , , on September 21, 2016 by xi'an

Rémi Bardenet and Adrien Hardy have arXived this paper a few months ago but I was a bit daunted by the sheer size of the paper, until I found the perfect opportunity last week..! The approach relates to probabilistic numerics as well as Monte Carlo, in that it can be seen as a stochastic version of Gaussian quadrature. The authors mention in the early pages a striking and recent result by Delyon and Portier that using an importance weight where the sampling density is replaced with the leave-one-out kernel estimate produces faster convergence than the regular Monte Carlo √n! Which reminds me of quasi-Monte Carlo of course, discussed in the following section (§1.3), with the interesting [and new to me] comment that the theoretical rate (and thus the improvement) does not occur until the sample size N is exponential in the dimension. Bardenet and Hardy achieve similar super-efficient convergence by mixing quadrature with repulsive simulation. For almost every integrable function.

The fact that determinantal point processes (on the unit hypercube) and Gaussian quadrature methods are connected is not that surprising once one considers that such processes are associated with densities made of determinants, which matrices are kernel-based, K(x,y), with K expressed as a sum of orthogonal polynomials. An N-th order determinantal process in dimension d satisfies a generalised Central Limit Theorem in that the speed of convergence is

\sqrt{N}^{(d-1)/d}

which means faster than √N…  This is more surprising, of course, even though one has to examine the conditions Continue reading

MCqMC 2016 [#4]

Posted in Mountains, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , , on August 21, 2016 by xi'an

In his plenary talk this morning, Arnaud Doucet discussed the application of pseudo-marginal techniques to the latent variable models he has been investigating for many years. And its limiting behaviour towards efficiency, with the idea of introducing correlation in the estimation of the likelihood ratio. Reducing complexity from O(T²) to O(T√T). With the very surprising conclusion that the correlation must go to 1 at a precise rate to get this reduction, since perfect correlation would induce a bias. A massive piece of work, indeed!

The next session of the morning was another instance of conflicting talks and I hoped from one room to the next to listen to Hani Doss’s empirical Bayes estimation with intractable constants (where maybe SAME could be of interest), Youssef Marzouk’s transport maps for MCMC, which sounds like an attractive idea provided the construction of the map remains manageable, and Paul Russel’s adaptive importance sampling that somehow sounded connected with our population Monte Carlo approach. (With the additional step of considering transform maps.)

An interesting item of information I got from the final announcements at MCqMC 2016 just before heading to Monash, Melbourne, is that MCqMC 2018 will take place in the city of Rennes, Brittany, on July 2-6. Not only it is a nice location on its own, but it is most conveniently located in space and time to attend ISBA 2018 in Edinburgh the week after! Just moving from one Celtic city to another Celtic city. Along with other planned satellite workshops, this occurrence should make ISBA 2018 more attractive [if need be!] for participants from oversea.

MCqMC 2016 [#2]

Posted in pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , , , , on August 17, 2016 by xi'an

In her plenary talk this morning, Christine Lemieux discussed connections between quasi-Monte Carlo and copulas, covering a question I have been considering for a while. Namely, when provided with a (multivariate) joint cdf F, is there a generic way to invert a vector of uniforms [or quasi-uniforms] into a simulation from F? For Archimedian copulas (as we always can get back to copulas), there is a resolution by the Marshall-Olkin representation,  but this puts a restriction on the distributions F that can be considered. The session on synthetic likelihoods [as introduced by Simon Wood in 2010] put together by Scott Sisson was completely focussed on using normal approximations for the distribution of the vector of summary statistics, rather than the standard ABC non-parametric approximation. While there is a clear (?) advantage in using a normal pseudo-likelihood, since it stabilises with much less simulations than a non-parametric version, I find it difficult to compare both approaches, as they lead to different posterior distributions. In particular, I wonder at the impact of the dimension of the summary statistics on the approximation, in the sense that it is less and less likely that the joint is normal as this dimension increases. Whether this is damaging for the resulting inference is another issue, possibly handled by a supplementary ABC step that would take the first-step estimate as summary statistic. (As a side remark, I am intrigued at everyone being so concerned with unbiasedness of methods that are approximations with no assessment of the amount of approximation!) The last session of the day was about multimodality and MCMC solutions, with talks by Hyungsuk Tak, Pierre Jacob and Babak Shababa, plus mine. Hunsuk presented the RAM algorithm I discussed earlier under the title of “love-hate” algorithm, which was a kind reference to my post! (I remain puzzled by the ability of the algorithm to jump to another mode, given that the intermediary step aims at a low or even zero probability region with an infinite mass target.) And Pierre talked about using SMC for Wang-Landau algorithms, with a twist to the classical stochastic optimisation schedule that preserves convergence. And a terrific illustration on a distribution inspired from the Golden Gate Bridge that reminded me of my recent crossing! The discussion around my folded Markov chain talk focussed on the extension of the partition to more than two sets, the difficulty being in generating automated projections, with comments about connections with computer graphic tools. (Too bad that the parallel session saw talks by Mark Huber and Rémi Bardenet that I missed! Enjoying a terrific Burmese dinner with Rémi, Pierre and other friends also meant I could not post this entry on time for the customary 00:16. Not that it matters in the least…)

Rémi Bardenet’s seminar

Posted in Kids, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , on April 7, 2016 by xi'an

Grand Palais from Esplanade des Invalides, Paris, Dec. 07, 2012Next week, Rémi Bardenet is giving a seminar in Paris, Thursday April 14, 2pm, in ENSAE [room 15] on MCMC methods for tall data. Unfortunately, I will miss this opportunity to discuss with Rémi as I will be heading to La Sapienza, Roma, for Clara Grazian‘s PhD defence the next day.  And on Monday afternoon, April 11, Nicolas Chopin will give a talk on quasi-Monte Carlo for sequential problems at Institut Henri Poincaré.