day one at ISBA 22

Started the day with a much appreciated swimming practice in the [alas warm⁺⁺⁺] outdoor 50m pool on the Island with no one but me in the slooow lane. And had my first ride with the biXi system, surprised at having to queue behind other bikes at red lights! More significantly, it was a great feeling to reunite at last with so many friends I had not met for more than two years!!!

My friend Adrian Raftery gave the very first plenary lecture on his work on the Bayesian approach to long-term population projections, which was recently  a work censored by some US States, then counter-censored by the Supreme Court [too busy to kill Roe v. Wade!]. Great to see the use of Bayesian methods validated by the UN Population Division [with at least one branch of the UN

Stephen Lauritzen returning to de Finetti notion of a model as something not real or true at all, back to exchangeability. Making me wonder when exchangeability is more than a convenient assumption leading to the Hewitt-Savage theorem. And sufficiency. I mean, without falling into a Keynesian fallacy, each point of the sample has unique specificities that cannot be taken into account in an exchangeable model. Nice to hear some measure theory, though!!! Plus a comment on the median never being sufficient, recouping an older (and presumably not original) point of mine. Stephen’s (or Fisher’s?) argument being that the median cannot be recursively computed!

Antonietta Mira and I had our ABC session this afternoon with Cecilia Viscardi, Sirio Legramanti, and Massimiliano Tamborino (Warwick) as speakers. Cecilia linked ABC with normalising flows, in collaboration with Dennis Prangle (whose earlier paper on this connection was presented as the first One World ABC seminar). Thus using past simulations to approximate the posterior by a neural network, possibly with a significant increase in computing time when compared with more rudimentary SMC-ABC methods in larger dimensions. Sirio considered summary-free ABC based on discrepancies like Rademacher complexity. Which more or less contains MMD, Kullback-Leibler, Wasserstein and more, although it seems to be dependent on the parameterisation of the observations. An interesting opening at the end was that this approach could apply to non iid settings. Massi presented a paper coauthored with Umberto that had just been arXived. On sequential ABC with a dependence on the summary statistic (hence guided). Further bringing copulas into the game, although this forces another choice [for the marginals] in the method.

Tamara Broderick talked about a puzzling leverage effect of some observations in economic studies where a tiny portion of individuals may modify the significance or the sign of a coefficient, for which I cannot tell whether the data or the reliance on statistical significance are to blame. Robert Kohn presented mixture-of-Gaussian copulas [not to be confused with mixture of Gaussian-copulas!] and Nancy Reid concluded my first [and somewhat exhausting!] day at ISBA with a BFF talk on the different statistical paradigms take on confidence (for which the notion of calibration seems to remain frequentist).

Side comments: First, most people in the conference are wearing masks, which is great! Also, I find it hard to read slides from the screen, which I presume is an age issue (?!) Even more aside, I had Korean lunch in a place that refused to serve me a glass of water, which I find amazing.

adaptive ABC tolerance

“There are three common approaches for selecting the tolerance sequence (…) [they] can lead to inefficient sampling”

Umberto Simola, Jessi Cisewski-Kehe, Michael Gutmann and Jukka Corander recently arXived a paper entitled Adaptive Approximate Bayesian Computation Tolerance Selection. I appreciate that they start from our ABC-PMC paper, i.e., Beaumont et al. (2009) [although the representation that the ABC tolerances are fixed in advance is somewhat incorrect in that we used in our codes quantiles of the distances to set our tolerances.] This is also the approach advocated for the initialisation step by the current paper.  Although remaining a wee bit vague. Subsequent steps are based on the proximity between the resulting approximations to the ABC posteriors, more exactly with a quantile derived from the maximum of the ratio between two estimated successive ABC posteriors. Mimicking the Accept-Reject step if always one step too late.  The iteration stops when the ratio is almost one, possibly missing the target due to Monte Carlo variability. (Recall that the “optimal” tolerance is not zero for a finite sample size.)

“…the decrease in the acceptance rate is mitigated by the improvement in the proposed particles.”

A problem is that it depends on the form of the approximation and requires non-parametric hence imprecise steps. Maybe variational encoders could help. Interesting approach by Sugiyama et al. (2012), of which I knew nothing, the core idea being that the ratio of two densities is also the solution to minimising a distance between the numerator density and a variable function times the bottom density. However since only the maximum of the ratio is needed, a more focused approach could be devised. Rather than first approximating the ratio and second maximising the estimated ratio. Maybe the solution of Goffinet et al. (1992) on estimating an accept-reject constant could work.

A further comment is that the estimated density is not properly normalised, which lessens the Accept-Reject analogy since the optimum may well stand above one. And thus stop “too soon”. (Incidentally, the paper contains the mixture example of Sisson et al. (2007), for which our own graphs were strongly criticised during our Biometrika submission!)

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.

optimal proposal for ABC

As pointed out by Ewan Cameron in a recent c’Og’ment, Justin Alsing, Benjamin Wandelt, and Stephen Feeney have arXived last August a paper where they discuss an optimal proposal density for ABC-SMC and ABC-PMC. Optimality being understood as maximising the effective sample size.

“Previous studies have sought kernels that are optimal in the (…) Kullback-Leibler divergence between the proposal KDE and the target density.”

The effective sample size for ABC-SMC is actually the regular ESS multiplied by the fraction of accepted simulations. Which surprisingly converges to the ratio

E[q(θ)/π(θ)|D]/E[π(θ)/q(θ)|D]

under the (true) posterior. (Where q(θ) is the importance density and π(θ) the prior density.] When optimised in q, this usually produces an implicit equation which results in a form of geometric mean between posterior and prior. The paper looks at approximate ways to find this optimum. Especially at an upper bound on q. Something I do not understand from the simulations is that the starting point seems to be the plain geometric mean between posterior and prior, in a setting where the posterior is supposedly unavailable… Actually the paper is silent on how the optimal can be approximated in practice, for the very reason I just mentioned. Apart from using a non-parametric or mixture estimate of the posterior after each SMC iteration, which may prove extremely costly when processed through the optimisation steps. However, an interesting if side outcome of these simulations is that the above geometric mean does much better than the posterior itself when considering the effective sample size.

astroABC: ABC SMC sampler for cosmological parameter estimation

“…the chosen statistic needs to be a so-called sufficient statistic in that any information about the parameter of interest which is contained in the data, is also contained in the summary statistic.”

Elise Jenningsa and Maeve Madigan arXived a paper on a new Python code they developed for implementing ABC-SMC, towards astronomy or rather cosmology applications. They stress the parallelisation abilities of their approach which leads to “crucial speed enhancement” against the available competitors, abcpmc and cosmoabc. The version of ABC implemented there is “our” ABC PMC where particle clouds are shifted according to mixtures of random walks, based on each and every point of the current cloud, with a scale equal to twice the estimated posterior variance. (The paper curiously refers to non-astronomy papers through their arXiv version, even when they have been published. Like our 2008 Biometrika paper.) A large part of the paper is dedicated to computing aspects that escape me, like the constant reference to MPIs. The algorithm is partly automated, except for the choice of the summary statistics and of the distance. The tolerance is chosen as a (large) quantile of the previous set of simulated distances. Getting comments from the designers of abcpmc and cosmoabc would be great.

“It is clear that the simple Gaussian Likelihood assumption in this case, which neglects the effects of systematics yields biased cosmological constraints.”

The last part of the paper compares ABC and MCMC on a supernova simulated dataset. Which is somewhat a dubious comparison since the model used for producing the data and running ABC is not the same as the Gaussian version used with MCMC. Unsurprisingly, MCMC then misses the true value of the cosmological parameters and most likely and more importantly the true posterior HPD region. While ABC SMC (or PMC) proceeds to a concentration around the genuine parameter values. (There is no additional demonstration of how accelerated the approach is.)