population quasi-Monte Carlo

“Population Monte Carlo (PMC) is an important class of Monte Carlo methods, which utilizes a population of proposals to generate weighted samples that approximate the target distribution”

A return of the prodigal son!, with this arXival by Huang, Joseph, and Mak, of a paper on population Monte Carlo using quasi-random sequences. The construct is based on an earlier notion of Joseph and Mak, support points, which are defined wrt a given target distribution F as minimising the variability of a sample from F away from these points. (I would have used instead my late friend Bernhard Flury’s principal points!) The proposal uses Owen-style scrambled Sobol points, followed by a deterministic mixture weighting à la PMC, followed by importance support resampling to find the next location parameters of the proposal mixture (which is why I included an unrelated mixture surface as my post picture!). This importance support resampling is obviously less variable than the more traditional ways of resampling but the cost moves from O(M) to O(M²).

“The main computational complexity of the algorithm is O(M²) from computing the pairwise distance of the M weighted samples”

The covariance parameters are updated as in our 2008 paper. This new proposal is interesting and reasonable, with apparent significant gains, albeit I would have liked to see a clearer discussion of the actual computing costs of PQMC.

distilling importance

As I was about to leave Warwick at the end of last week, I noticed a new arXival by Dennis Prangle, distilling importance sampling. In connection with [our version of] population Monte Carlo, “each step of [Dennis’] distilled importance sampling method aims to reduce the Kullback Leibler (KL) divergence from the distilled density to the current tempered posterior.”  (The introduction of the paper points out various connections with ABC, conditional density estimation, adaptive importance sampling, X entropy, &tc.)

“An advantage of [distilled importance sampling] over [likelihood-free] methods is that it performs inference on the full data, without losing information by using summary statistics.”

A notion used therein I had not heard before is the one of normalising flows, apparently more common in machine learning and in particular with GANs. (The slide below is from Shakir Mohamed and Danilo Rezende.) The  notion is to represent an arbitrary variable as the bijective transform of a standard variate like a N(0,1) variable or a U(0,1) variable (calling the inverse cdf transform). The only link I can think of is perfect sampling where the representation of all simulations as a function of a white noise vector helps with coupling.

I read a blog entry by Eric Jang on the topic (who produced this slide among other things) but did not emerge much the wiser. As the text instantaneously moves from the Jacobian formula to TensorFlow code… In Dennis’ paper, it appears that the concept is appealing for quickly producing samples and providing a rich family of approximations, especially when neural networks are included as transforms. They are used to substitute for a tempered version of the posterior target, validated as importance functions and aiming at being the closest to this target in Kullback-Leibler divergence. With the importance function interpretation, unbiased estimators of the gradient [in the parameter of the normalising flow] can be derived, with potential variance reduction. What became clearer to me from reading the illustration section is that the prior x predictive joint can also be modeled this way towards producing reference tables for ABC (or GANs) much faster than with the exact model. (I came across several proposals of that kind in the past months.) However, I deem mileage should vary depending on the size and dimension of the data. I also wonder at the connection between the (final) distribution simulated by distilled importance [the least tempered target?] and the ABC equivalent.

revisiting the balance heuristic

Last August, Felipe Medina-Aguayo (a former student at Warwick) and Richard Everitt (who has now joined Warwick) arXived a paper on multiple importance sampling (for normalising constants) that goes “exploring some improvements and variations of the balance heuristic via a novel extended-space representation of the estimator, leading to straightforward annealing schemes for variance reduction purposes”, with the interesting side remark that Rao-Blackwellisation may prove sub-optimal when there are many terms in the proposal family, in the sense that not every term in the mixture gets sampled. As already noticed by Victor Elvira and co-authors, getting rid of the components that are not used being an improvement without inducing a bias. The paper also notices that the loss due to using sample sizes rather than expected sample sizes is of second order, compared with the variance of the compared estimators. It further relates to a completion or auxiliary perspective that reminds me of the approaches we adopted in the population Monte Carlo papers and in the vanilla Rao-Blackwellisation paper. But it somewhat diverges from this literature when entering a simulated annealing perspective, in that the importance distributions it considers are freely chosen as powers of a generic target. It is quite surprising that, despite the normalising weights being unknown, a simulated annealing approach produces an unbiased estimator of the initial normalising constant. While another surprise therein is that the extended target associated to their balance heuristic does not admit the right density as marginal but preserves the same normalising constant… (This paper will be presented at BayesComp 2020.)

optimal choice among MCMC kernels

Last week in Siem Reap, Florian Maire [who I discovered originates from a Norman town less than 10km from my hometown!] presented an arXived joint work with Pierre Vandekerkhove at the Data Science & Finance conference in Cambodia that considers the following problem: Given a large collection of MCMC kernels, how to pick the best one and how to define what best means. Going by mixtures is a default exploration of the collection, as shown in (Tierney) 1994 for instance since this improves on both kernels (esp. when each kernel is not irreducible on its own!). This paper considers a move to local weights in the mixture, weights that are not estimated from earlier simulations, contrary to what I first understood.

As made clearer in the paper the focus is on filamentary distributions that are concentrated nearby lower-dimension sets or manifolds Since then the components of the kernel collections can be restricted to directions of these manifolds… Including an interesting case of a 2-D highly peaked target where converging means mostly simulating in x¹ and covering the target means mostly simulating in x². Exhibiting a schizophrenic tension between the two goals. Weight locally dependent means correction by Metropolis step, with cost O(n). What of Rao-Blackwellisation of these mixture weights, from weight x transition to full mixture, as in our PMC paper? Unclear to me as well [during the talk] is the use in the mixture of basic Metropolis kernels, which are not absolutely continuous, because of the Dirac mass component. But this is clarified by Section 5 in the paper. A surprising result from the paper (Corollary 1) is that the use of local weights ω(i,x) that depend on the current value of the chain does jeopardize the stationary measure π(.) of the mixture chain. Which may be due to the fact that all components of the mixture are already π-invariant. Or that the index of the kernel constitutes an auxiliary (if ancillary)  variate. (Algorithm 1 in the paper reminds me of delayed acceptance. Making me wonder if computing time should be accounted for.) A final question I briefly discussed with Florian is the extension to weights that are automatically constructed from the simulations and the target.

ABC by QMC

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.