**T**oday, I read a newly arXived paper by Stephen Gratton on a method called GLASS for *General Likelihood Approximate Solution Scheme*… The starting point is the same as with ABC or synthetic likelihood, namely a collection of summary statistics and an intractable likelihood. The author proposes to use as a substitute a maximum entropy solution based on these summary statistics and their assumed moments under the theoretical model. What is quite unclear in the paper is whether or not these assumed moments are available in closed form or not. Otherwise, it would appear as a variant to the synthetic likelihood [aka simulated moments] approach, meaning that the expectations of the summary statistics under the theoretical model and for a given value of the parameter are obtained through Monte Carlo approximations. (All the examples therein allow for closed form expressions.)

## Archive for synthetic likelihood

## approximate likelihood

Posted in Books, Statistics with tags ABC, arXiv, likelihood-free methods, maximum entropy, synthetic likelihood, untractable normalizing constant on September 6, 2017 by xi'an## ABC’ory in Banff [17w5025]

Posted in Mountains, pictures, Statistics, Travel, University life with tags ABC, ABC convergence, Banff, Banff International Research Station for Mathematical Innovation, BIRS, Canada, Canadian Rockies, consistency of ABC methods, coverage, synthetic likelihood on February 23, 2017 by xi'an**A**nother great day of talks and discussions at BIRS! Continuing on the themes of the workshop between delving into the further validation of those approximation techniques and the devising of ever more approximate solutions for ever more complex problems. Among the points that came clearer to me through discussion, a realisation that the synthetic likelihood perspective is not that far away from our assumptions in the consistency paper. And that a logistic version of the approach can be constructed as well. A notion I had not met before (or have forgotten I had met) is the one of early rejection ABC, which should actually be investigated more thoroughly as it should bring considerable improvement in computing time (with the caveats of calibrating the acceptance step before producing the learning sample and of characterising the output). Both Jukka Corander and Ewan Cameron reminded us of the case of models that take minutes or hours to produce one single dataset. (In his talk on some challenging applications, Jukka Corander chose to move from socks to boots!) And Jean-Michel Marin produced an illuminating if sobering experiment on the lack of proper Bayesian coverage by ABC solutions. (It appears that Ewan’s video includes a long empty moment when we went out for the traditional group photo, missing the end of his talk.)

## ABC’ory in Banff [17w5025]

Posted in Mountains, pictures, Statistics, Travel, University life with tags 17w5025, ABC, Approximate Bayesian computation, Banff, BIRS, Canada, convergence, Les Diablerets, Rocky Mountains, synthetic likelihood on February 21, 2017 by xi'an**T**he ABC workshop I co-organised has now started and, despite a few last minutes cancellations, we have gathered a great crowd of researchers on the validation and expansion of ABC methods. Or ABC’ory to keep up with my naming of workshops. The videos of the talks should come up progressively on the BIRS webpage. When I did not forget to launch the recording. The program is quite open and with this size of workshop allows for talks and discussions to last longer than planned: the first days contain several expository talks on ABC convergence, auxiliary or synthetic models, summary constructions, challenging applications, dynamic models, and model assessment. Plus prepared discussions on those topics that hopefully involve several workshop participants. We had also set some time for snap-talks, to induce everyone to give a quick presentation of one’s on-going research and open problems. The first day was rather full but saw a lot of interactions and discussions during and around the talks, a mood I hope will last till Friday! Today in replacement of Richard Everitt who alas got sick just before the workshop, we are conducting a discussion on dimensional issues, part of which is made of parts of the following slides (mostly recycled from earlier talks, including the mini-course in Les Diablerets):

## MCqMC 2016 [#2]

Posted in pictures, Running, Statistics, Travel, University life with tags California, Cambodia, conference, copulas, folded Markov chain, Golden Gate Bridge, Laplace-Stieljes transform, love-hate algorithm, Marshall-Olkin algorithm, MCMC, MCqMC 2016, Monte Carlo Statistical Methods, quadrangle, quasi-Monte Carlo methods, simulation, SMC, Stanford University, summary statistics, synthetic likelihood, The night of the hunter, Wang-Landau algorithm on August 17, 2016 by xi'anIn 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…)

## can we trust computer simulations? [day #2]

Posted in Books, pictures, Statistics, Travel, University life with tags ABC, chaos, climate, climatic-skeptics, design of experiments, ecological models, Ecology, epistemology, equivalence test, ergodicity, Germany, Hannover, Herrenhausen Palace, manufactured solutions, philosophy of sciences, reproducible research, simulation, sociology, state space model, synthetic likelihood, validation, verification, WYSINWYG on July 13, 2015 by xi'an*“Sometimes the models are better than the data.” G. Krinner*

**S**econd day at the conference on building trust in computer simulations. Starting with a highly debated issue, climate change projections. Since so many criticisms are addressed to climate models as being not only wrong but also unverifiable. And uncheckable. As explained by Gerhart Krinner, the IPCC has developed methodologies to compare models and evaluate predictions. However, from what I understood, this validation does not say anything about the future, which is the part of the predictions that matters. And that is attacked by critics and feeds climatic-skeptics. Because it is so easy to argue against the homogeneity of the climate evolution and for “*what you’ve seen is not what you’ll get*“! (Even though climatic-skeptics are the least likely to use this time-heterogeneity argument, being convinced as they are of the lack of human impact over the climate.) The second talk was by Viktoria Radchuk about validation in ecology. Defined here as a test of predictions against independent data (and designs). And mentioning Simon Wood’s synthetic likelihood as the Bayesian reference for conducting model choice (as a synthetic likelihoods ratio). I had never thought of this use (found in Wood’s original paper) for synthetic likelihood, I feel a bit queasy about using a synthetic likelihood ratio as a genuine likelihood ratio. Which led to a lively discussion at the end of her talk. The next talk was about validation in economics by Matteo Richiardi, who discussed state-space models where the hidden state is observed through a summary statistic, perfect playground for ABC! But Matteo opted instead for a non-parametric approach that seems to increase imprecision and that I have never seen used in state-space models. The last part of the talk was about non-ergodic models, for which checking for validity becomes much more problematic, in my opinion. Unless one manages multiple observations of the non-ergodic path. Nicole Saam concluded this “Validation in…” morning with Validation in Sociology. With a more pessimistic approach to the possibility of finding a falsifying strategy, because of the vague nature of sociology models. For which data can never be fully informative. She illustrated the issue with an EU negotiation analysis. Where most hypotheses could hardly be tested.

“Bayesians persist with poor examples of randomness.” L. Smith

“Bayesians can be extremely reasonable.” L. Smith

The afternoon session was dedicated to methodology, mostly statistics! Andrew Robinson started with a talk on (frequentist) model validation. Called splitters and lumpers. Illustrated by a forest growth model. He went through traditional hypothesis tests like Neyman-Pearson’s that try to split between samples. And (bio)equivalence tests that take difference as the null. Using his equivalence R package. Then Leonard Smith took over [in a literal way!] from a sort-of-Bayesian perspective, in a work joint with Jim Berger and Gary Rosner on pragmatic Bayes which was mostly negative about Bayesian modelling. Introducing (to me) the compelling notion of structural model error as a representation of the inadequacy of the model. With illustrations from weather and climate models. His criticism of the Bayesian approach is that it cannot be holistic while pretending to be [my wording]. And being inadequate to measure model inadequacy, to the point of making prior choice meaningless. Funny enough, he went back to the ball dropping experiment David Higdon discussed at one JSM I attended a while ago, with the unexpected outcome that one ball did not make it to the bottom of the shaft. A more positive side was that posteriors are useful models but should not be interpreted from a probabilistic perspective. Move beyond probability was his final message. (For most of the talk, I misunderstood P(BS), the probability of a big surprise, for something else…) This was certainly the most provocative talk of the conference and the discussion could have gone on for the rest of day! Somewhat, Lenny was voluntarily provocative in piling the responsibility upon the Bayesian’s head for being overconfident and not accounting for the physicist’ limitations in modelling the phenomenon of interest. Next talk was by Edward Dougherty on methods used in biology. He separated within-model uncertainty from outside-model inadequacy. The within model part is mostly easy to agree upon. Even though difficulties in estimating parameters creates uncertainty classes of models. Especially because of being from a small data discipline. He analysed the impact of machine learning techniques like classification as being useless without prior knowledge. And argued in favour of the Bayesian minimum mean square error estimator. Which can also lead to a classifier. And experimental design. (Using MSE seems rather reductive when facing large dimensional parameters.) Last talk of the day was by Nicolas Becu, a geographer, with a surprising approach to validation via stakeholders. A priori not too enticing a name! The discussion was of a more philosophical nature, going back to (re)define validation against reality and imperfect models. And including social aspects of validation, e.g., reality being socially constructed. This led to the stakeholders, because a model is then a shared representation. Nicolas illustrated the construction by simulation “games” of a collective model in a community of Thai farmers and in a group of water users.

In a rather unique fashion, we also had an evening discussion on points we share and points we disagreed upon. After dinner (and wine), which did not help I fear! Bill Oberkampf mentioned the use of manufactured solutions to check code, which seemed very much related to physics. But then we got mired into the necessity of dividing between verification and validation. Which sounded very and too much engineering-like to me. Maybe because I do not usually integrate coding errors and algorithmic errors into my reasoning (verification)… Although sharing code and making it available makes a big difference. Or maybe because considering *all* models are wrong is neither part of my methodology (validation). This part ended up in a fairly pessimistic conclusion on the lack of trust in most published articles. At least in the biological sciences.

## Hamiltonian ABC

Posted in Books, pictures, Statistics, University life with tags ABC, Amsterdam, Hamiltonian Monte Carlo, Markov chain, Monte Carlo Statistical Methods, pseudo-random generator, random seed, synthetic likelihood on March 13, 2015 by xi'an**O**n Monday, Ed Meeds, Robert Leenders, and Max Welling (from Amsterdam) arXived a paper entitled Hamiltonian ABC. Before looking at the paper in any detail, I got puzzled by this association of antagonistic terms, since ABC is intended for complex and mostly intractable likelihoods, while Hamiltonian Monte Carlo requires a lot from the target, in order to compute gradients and Hessians… *[Warning: some graphs on pages 13-14 may be harmful to your printer!]*

Somewhat obviously (ex-post!), the paper suggests to use Hamiltonian dynamics on ABC approximations of the likelihood. They compare a Gaussian kernel version

with the synthetic Gaussian likelihood version of Wood (2010)

where both mean and variance are estimated from the simulated data. If ε is taken as an external quantity and driven to zero, the second approach is much more stable. But… ε is never driven to zero in ABC, or fixed at ε=0.37: It is instead considered as a kernel bandwidth and hence estimated from the simulated data. Hence ε is commensurable with σ(θ). And this makes me wonder at the relevance of the conclusion that synthetic is better than kernel for Hamiltonian ABC. More globally, I wonder at the relevance of better simulating from a still approximate target when the true goal is to better approximate the genuine posterior.

Some of the paper covers separate issues like handling gradient by finite differences à la Spall *[if you can afford it!]* and incorporating the random generator as part of the Markov chain. And using S *common* random numbers in computing the gradients for all values of θ. (Although I am not certain all random generators can be represented as a deterministic transform of a parameter θ and of a fixed number of random uniforms. But the authors may consider a random number of random uniforms when they represent their random generators as deterministic transform of a parameter θ and of the random seed. I am also uncertain about the distinction between common, sticky, and persistent random numbers!)

## Inference for stochastic simulation models by ABC

Posted in Books, Statistics, University life with tags ABC, ABC validation, Bayesian optimisation, non-parametrics, sufficiency, synthetic likelihood on February 13, 2015 by xi'an**H**artig et al. published a while ago (2011) a paper in *Ecology Letters* entitled “Statistical inference for stochastic simulation models – theory and application”, which is mostly about ABC. (Florian Hartig pointed out the paper to me in a recent blog comment. about my discussion of the early parts of Guttman and Corander’s paper.) The paper is largely a tutorial and it reminds the reader about related methods like indirect inference and methods of moments. The authors also insist on presenting ABC as a particular case of likelihood approximation, whether non-parametric or parametric. Making connections with pseudo-likelihood and pseudo-marginal approaches. And including a discussion of the possible misfit of the assumed model, handled by an external error model. And also introducing the notion of *informal likelihood* (which could have been nicely linked with *empirical likelihood*). A last class of approximations presented therein is called *rejection filters* and reminds me very much of Ollie Ratman’s papers.

“Our general aim is to find sufficient statistics that are as close to minimal sufficiency as possible.” (p.819)

As in other ABC papers, and as often reported on this blog, I find the stress on sufficiency a wee bit too heavy as those models calling for approximation almost invariably do not allow for any form of useful sufficiency. Hence the mathematical statistics notion of sufficiency is mostly useless in such settings.

“A basic requirement is that the expectation value of the point-wise approximation of p(S^{obs}|φ) must be unbiased” (p.823)

As stated above the paper is mostly in tutorial mode, for instance explaining what MCMC and SMC methods are. As illustrated by the above figure. There is however a final and interesting discussion section on the impact of estimating the likelihood function at different values of the parameter. However, the authors seem to focus solely on pseudo-marginal results to validate this approximation, hence on unbiasedness, which does not work for most ABC approaches that I know. And for the approximations listed in the survey. Actually, it would be quite beneficial to devise a cheap tool to assess the bias or extra-variation due to the use of approximative techniques like ABC… A sort of 21st Century bootstrap?!