Archive for ABC-SMC

adaptive copulas for ABC

Posted in Statistics with tags , , , , , , , , on March 20, 2019 by xi'an

A paper on ABC I read on my way back from Cambodia:  Yanzhi Chen and Michael Gutmann arXived an ABC [in Edinburgh] paper on learning the target via Gaussian copulas, to be presented at AISTATS this year (in Okinawa!). Linking post-processing (regression) ABC and sequential ABC. The drawback in the regression approach is that the correction often relies on an homogeneity assumption on the distribution of the noise or residual since this approach only applies a drift to the original simulated sample. Their method is based on two stages, a coarse-grained one where the posterior is approximated by ordinary linear regression ABC. And a fine-grained one, which uses the above coarse Gaussian version as a proposal and returns a Gaussian copula estimate of the posterior. This proposal is somewhat similar to the neural network approach of Papamakarios and Murray (2016). And to the Gaussian copula version of Li et al. (2017). The major difference being the presence of two stages. The new method is compared with other ABC proposals at a fixed simulation cost, which does not account for the construction costs, although they should be relatively negligible. To compare these ABC avatars, the authors use a symmetrised Kullback-Leibler divergence I had not met previously, requiring a massive numerical integration (although this is not an issue for the practical implementation of the method, which only calls for the construction of the neural network(s)). Note also that sequential ABC is only run for two iterations, and also that none of the importance sampling ABC versions of Fearnhead and Prangle (2012) and of Li and Fearnhead (2018) are considered, all versions relying on the same vector of summary statistics with a dimension much larger than the dimension of the parameter. Except in our MA(2) example, where regression does as well. I wonder at the impact of the dimension of the summary statistic on the performances of the neural network, i.e., whether or not it is able to manage the curse of dimensionality by ignoring all but essentially the data  statistics in the optimisation.

particular degeneracy in ABC model choice

Posted in Statistics with tags , , , , , , on February 22, 2019 by xi'an

In one of the presentations by the last cohort of OxWaSP students, the group decided to implement an ABC model choice strategy based on sequential ABC inspired from Toni et al.  (2008). and this made me reconsider this approach (disclaimer: no criticism of the students implied in the following!). Indeed, the outcome of the simulation led to the ultimate selection of a single model, exclusive of all other models, corresponding to a posterior probability of one in favour of this model. Which sounds like a drawback of the ABC-SMC model choice approach in this setting, namely that it is quite prone to degeneracy, much more than standard SMC, since once a model vanishes from the list, it can never reappear in the following iterations if I am reading the algorithm correctly. To avoid this degeneracy, one would need to keep a population of particles of a given size, for each model, towards using it as a pool for moves at following iterations… Which also means that running in parallel as many ABC-SMC filters as there are models would be equally or more efficient, a wee bit like parallel MCMC chains may prove more efficient than reversible jump for model comparison. (On the trivial side, the OxWaSP seminar on the same day was briefly interrupted by water leakage caused by Storm Eric and poor workmanship on the new building!)

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.

optimal proposal for ABC

Posted in Statistics with tags , , , , , , , , , , on October 8, 2018 by xi'an

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.

ABC with no prior

Posted in Books, Kids, pictures with tags , , , , , , on April 30, 2018 by xi'an

“I’m trying to fit a complex model to some data that take a large amount of time to run. I’m also unable to write down a Likelihood function to this problem and so I turned to approximate Bayesian computation (ABC). Now, given the slowness of my simulations, I used Sequential ABC (…) In fact, contrary to the concept of Bayesian statistics (new knowledge updating old knowledge) I would like to remove all the influence of the priors from my estimates. “

A question from X validated where I have little to contribute as the originator of the problem had the uttermost difficulties to understand that ABC could not be run without a probability structure on the parameter space. Maybe a fiducialist in disguise?! To this purpose this person simulated from a collection of priors and took the best 5% across the priors, which is akin to either running a mixture prior or to use ABC for conducting prior choice, which reminds me of a paper of Toni et al. Not that it helps removing “all the influence of the priors”, of course…

An unrelated item of uninteresting trivia is that a question I posted in 2012 on behalf of my former student Gholamossein Gholami about the possibility to use EM to derive a Weibull maximum likelihood estimator (instead of sheer numerical optimisation) got over the 10⁴ views. But no answer so far!

delayed acceptance ABC-SMC

Posted in pictures, Statistics, Travel with tags , , , , , , , on December 11, 2017 by xi'an

Last summer, during my vacation on Skye,  Richard Everitt and Paulina Rowińska arXived a paper on delayed acceptance associated with ABC. ArXival that I missed, then! In order to decrease the number of simulations from the likelihood. As in our own delayed acceptance paper (without ABC), a cheap alternative generator is used to first reject the least likely parameters values, before possibly continuing to use a full generator. Also as lazy ABC. The first step of this ABC algorithm requires a cheap generator plus a primary tolerance ε¹ to compare the generation with the data or part of it. This may be followed by a second generation with a second tolerance level ε². The paper applies more specifically ABC-SMC as introduced in Sisson, Fan and Tanaka (2007) and reassessed in our subsequent 2009 Biometrika paper with Mark Beaumont, Jean-Marie Cornuet and Jean-Michel Marin. As well as in the ABC-SMC paper by Pierre Del Moral and Arnaud Doucet.

When looking at the version of the algorithm [Algorithm 2] based on two basic acceptance ABC steps, there are two features I find intriguing: (i) the primary step uses a cheap generator to reject early poor values of the parameter, followed by the second step involving a more expensive and exact generator, but I see no impact of the choice of this cheap generator in the acceptance probability; (ii) this is an SMC algorithm with imposed resampling at each iteration but there is no visible step for creating new weights after the resampling step. In the current presentation, it sounds like the weights do not change from the initial step, except for those turning to zero and the renormalisation transforms. Which makes the (unspecified) stratification of little interest if any. I must therefore miss a point in the implementation!

One puzzling sentence in the appendix is that the resampling algorithm used in the SMC step “ensures that every particle that is alive before resampling is represented in the resampled particles”, which reminds me of an argument [possibly a different one] made already in Sisson, Fan and Tanaka (2007) and that we could not validate in our subsequent paper. For resampling to be correct, a form of multinomial sampling must be implemented, even via variance reduction schemes like stratified or systematic sampling.

probably ABC [and provably robust]

Posted in Books, pictures, Statistics, Travel with tags , , , , , , , , on August 8, 2017 by xi'an

Two weeks ago, James Ridgway (formerly CREST) arXived a paper on misspecification and ABC, a topic on which David Frazier, Judith Rousseau and I have been working for a while now [and soon to be arXived as well].  Paper that I re-read on a flight to Amsterdam [hence the above picture], written as a continuation of our earlier paper with David, Gael, and Judith. One specificity of the paper is to use an exponential distribution on the distance between the observed and simulated sample within the ABC distribution. Which reminds me of the resolution by Bissiri, Holmes, and Walker (2016) of the intractability of the likelihood function. James’ paper contains oracle inequalities between the ABC approximation and the genuine distribution of the summary statistics, like a bound on the distance between the expectations of the summary statistics under both models. Which writes down as a sum of a model bias, of two divergences between empirical and theoretical averages, on smoothness penalties, and on a prior impact term. And a similar bound on the distance between the expected distance to the oracle estimator of θ under the ABC distribution [and a Lipschitz type assumption also found in our paper]. Which first sounded weird [to me] as I would have expected the true posterior, until it dawned on me that the ABC distribution is the one used for the estimation [a passing strike of over-Bayesianism!]. While the oracle bound could have been used directly to discuss the rate of convergence of the exponential rate λ to zero [with the sample size n], James goes into the interesting alternative direction of setting a prior on λ, an idea that dates back to Olivier Catoni and Peter Grünwald. Or rather a pseudo-posterior on λ, a common occurrence in the PAC-Bayesian literature. In one of his results, James obtains a dependence of λ on the dimension m of the summary [as well as the root dependence on the sample size n], which seems to contradict our earlier independence result, until one realises this scale parameter is associated with a distance variable, itself scaled in m.

The paper also contains a non-parametric part, where the parameter θ is the unknown distribution of the data and the summary the data itself. Which is quite surprising as I did not deem it possible to handle non-parametrics with ABC. Especially in a misspecified setting (although I have trouble perceiving what this really means).

“We can use most of the Monte Carlo toolbox available in this context.”

The theoretical parts are a bit heavy on notations and hard to read [as a vacation morning read at least!]. They are followed by a Monte Carlo implementation using SMC-ABC.  And pseudo-marginals [at least formally as I do not see how the specific features of pseudo-marginals are more that an augmented representation here]. And adaptive multiple pseudo-samples that reminded me of the Biometrika paper of Anthony Lee and Krys Latuszynski (Warwick). Therefore using indeed most of the toolbox!