Archive for ABC

astronomical evidence

Posted in pictures, Statistics, University life with tags , , , , , , , , , , , , on July 24, 2015 by xi'an

As I have a huge arXiv backlog and an even higher non-arXiv backlog, I cannot be certain I will find time to comment on those three recent and quite exciting postings connecting ABC with astro- and cosmo-statistics [thanks to Ewan for pointing out those to me!]:

Measuring statistical evidence using relative belief [book review]

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , , , , , , on July 22, 2015 by xi'an

“It is necessary to be vigilant to ensure that attempts to be mathematically general do not lead us to introduce absurdities into discussions of inference.” (p.8)

This new book by Michael Evans (Toronto) summarises his views on statistical evidence (expanded in a large number of papers), which are a quite unique mix of Bayesian  principles and less-Bayesian methodologies. I am quite glad I could receive a version of the book before it was published by CRC Press, thanks to Rob Carver (and Keith O’Rourke for warning me about it). [Warning: this is a rather long review and post, so readers may chose to opt out now!]

“The Bayes factor does not behave appropriately as a measure of belief, but it does behave appropriately as a measure of evidence.” (p.87)

Continue reading

MCMskv, Lenzerheide, 4-7 Jan., 2016 [news #1]

Posted in Kids, Mountains, pictures, R, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , , on July 20, 2015 by xi'an

moonriseThe BayesComp MCMski V [or MCMskv for short] has now its official website, once again maintained by Merrill Lietchy from Drexel University, Philadelphia, and registration is even open! The call for contributed sessions is now over, while the call for posters remains open until the very end. The novelty from the previous post is that there will be a “Breaking news” [in-between the Late news sessions at JSM and the crash poster talks at machine-learning conferences] session to highlight major advances among poster submissions. And that there will be an opening talk by Steve [the Bayesian] Scott on the 4th, about the frightening prospect of MCMC death!, followed by a round-table and a welcome reception, sponsored by the Swiss Supercomputing Centre. Hence the change in dates. Which still allows for arrivals in Zürich on the January 4th [be with you].

Leave the Pima Indians alone!

Posted in Books, R, Statistics, University life with tags , , , , , , , , , , , , , , , , , on July 15, 2015 by xi'an

“…our findings shall lead to us be critical of certain current practices. Specifically, most papers seem content with comparing some new algorithm with Gibbs sampling, on a few small datasets, such as the well-known Pima Indians diabetes dataset (8 covariates). But we shall see that, for such datasets, approaches that are even more basic than Gibbs sampling are actually hard to beat. In other words, datasets considered in the literature may be too toy-like to be used as a relevant benchmark. On the other hand, if ones considers larger datasets (with say 100 covariates), then not so many approaches seem to remain competitive” (p.1)

Nicolas Chopin and James Ridgway (CREST, Paris) completed and arXived a paper they had “threatened” to publish for a while now, namely why using the Pima Indian R logistic or probit regression benchmark for checking a computational algorithm is not such a great idea! Given that I am definitely guilty of such a sin (in papers not reported in the survey), I was quite eager to read the reasons why! Beyond the debate on the worth of such a benchmark, the paper considers a wider perspective as to how Bayesian computation algorithms should be compared, including the murky waters of CPU time versus designer or programmer time. Which plays against most MCMC sampler.

As a first entry, Nicolas and James point out that the MAP can be derived by standard a Newton-Raphson algorithm when the prior is Gaussian, and even when the prior is Cauchy as it seems most datasets allow for Newton-Raphson convergence. As well as the Hessian. We actually took advantage of this property in our comparison of evidence approximations published in the Festschrift for Jim Berger. Where we also noticed the awesome performances of an importance sampler based on the Gaussian or Laplace approximation. The authors call this proposal their gold standard. Because they also find it hard to beat. They also pursue this approximation to its logical (?) end by proposing an evidence approximation based on the above and Chib’s formula. Two close approximations are provided by INLA for posterior marginals and by a Laplace-EM for a Cauchy prior. Unsurprisingly, the expectation-propagation (EP) approach is also implemented. What EP lacks in theoretical backup, it seems to recover in sheer precision (in the examples analysed in the paper). And unsurprisingly as well the paper includes a randomised quasi-Monte Carlo version of the Gaussian importance sampler. (The authors report that “the improvement brought by RQMC varies strongly across datasets” without elaborating for the reasons behind this variability. They also do not report the CPU time of the IS-QMC, maybe identical to the one for the regular importance sampling.) Maybe more surprising is the absence of a nested sampling version.

pimcisIn the Markov chain Monte Carlo solutions, Nicolas and James compare Gibbs, Metropolis-Hastings, Hamiltonian Monte Carlo, and NUTS. Plus a tempering SMC, All of which are outperformed by importance sampling for small enough datasets. But get back to competing grounds for large enough ones, since importance sampling then fails.

“…let’s all refrain from now on from using datasets and models that are too simple to serve as a reasonable benchmark.” (p.25)

This is a very nice survey on the theme of binary data (more than on the comparison of algorithms in that the authors do not really take into account design and complexity, but resort to MSEs versus CPus). I however do not agree with their overall message to leave the Pima Indians alone. Or at least not for the reason provided therein, namely that faster and more accurate approximations methods are available and cannot be beaten. Benchmarks always have the limitation of “what you get is what you see”, i.e., the output associated with a single dataset that only has that many idiosyncrasies. Plus, the closeness to a perfect normal posterior makes the logistic posterior too regular to pause a real challenge (even though MCMC algorithms are as usual slower than iid sampling). But having faster and more precise resolutions should on the opposite be  cause for cheers, as this provides a reference value, a golden standard, to check against. In a sense, for every Monte Carlo method, there is a much better answer, namely the exact value of the integral or of the optimum! And one is hardly aiming at a more precise inference for the benchmark itself: those Pima Indians [whose actual name is Akimel O’odham] with diabetes involved in the original study are definitely beyond help from statisticians and the model is unlikely to carry out to current populations. When the goal is to compare methods, as in our 2009 paper for Jim Berger’s 60th birthday, what matters is relative speed and relative ease of implementation (besides the obvious convergence to the proper target). In that sense bigger and larger is not always relevant. Unless one tackles really big or really large datasets, for which there is neither benchmark method nor reference value.

SPA 2015 Oxford

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , on July 14, 2015 by xi'an

Today I gave a talk on Approximate Bayesian model choice via random forests at the yearly SPA (Stochastic Processes and their Applications) 2015 conference, taking place in Oxford (a nice town near Warwick) this year. In Keble College more precisely. The slides are below and while they are mostly repetitions of earlier slides, there is a not inconsequential novelty in the presentation, namely that I included our most recent and current perspective on ABC model choice. Indeed, when travelling to Montpellier two weeks ago, we realised that there was a way to solve our posterior probability conundrum!

campusDespite the heat wave that rolled all over France that week, we indeed figured out a way to estimate the posterior probability of the selected (MAP) model, way that we had deemed beyond our reach in previous versions of the talk and of the paper. The fact that we could not provide an estimate of this posterior probability and had to rely instead on a posterior expected loss was one of the arguments used by the PNAS reviewers in rejecting the paper. While the posterior expected loss remains a quantity worth approximating and reporting, the idea that stemmed from meeting together in Montpellier is that (i) the posterior probability of the MAP is actually related to another posterior loss, when conditioning on the observed summary statistics and (ii) this loss can be itself estimated via a random forest, since it is another function of the summary statistics. A posteriori, this sounds trivial but we had to have a new look at the problem to realise that using ABC samples was not the only way to produce an estimate of the posterior probability! (We are now working on the revision of the paper for resubmission within a few week… Hopefully before JSM!)

can we trust computer simulations? [day #2]

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , , , , , , on July 13, 2015 by xi'an

Herrenhausen“Sometimes the models are better than the data.” G. Krinner

Second 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.

analysing statistical and computational trade-off of estimation procedures

Posted in Books, pictures, Statistics, University life with tags , , , , , , on July 8, 2015 by xi'an

bostown1

“The collection of estimates may be determined by questions such as: How much storage is available? Can all the data be kept in memory or only a subset? How much processing power is available? Are there parallel or distributed systems that can be exploited?”

Daniel Sussman, Alexander Volfovsky, and Edoardo Airoldi from Harvard wrote a very interesting paper about setting a balance between statistical efficiency and computational efficiency, a theme that resonates with our recent work on ABC and older considerations about the efficiency of Monte Carlo algorithms. While the paper avoids drifting towards computer science even with a notion like algorithmic complexity, I like the introduction of a loss function in the comparison game, even though the way to combine both dimensions is unclear. And may limit the exercise to an intellectual game. In an ideal setting one would set the computational time, like “I have one hour to get this estimate”, and compare risks under that that computing constraint. Possibly dumping some observations from the sample to satisfy the constraint. Ideally. Which is why this also reminds me of ABC: given an intractable likelihood, one starts by throwing away some data precision by using a tolerance ε and usually more through an insufficient statistic. Hence ABC procedures could also be compared in such terms.

In the current paper, the authors only compare schemes of breaking the sample into bits to handle each observation only once. Meaning it cannot be used in both the empirical mean and the empirical variance. This sounds a bit contrived in that the optimum allocation depends on the value of the parameter the procedure attempts to estimate. Still, it could lead to a new form of bandit problems: given a bandit with as many arms as there are parameters, at each new observation, decide on the allocation towards minimising the overall risk. (There is a missing sentence at the end of Section 4.)

Any direction for turning those considerations into a practical decision machine would be fantastic, although the difficulties are formidable, from deciding between estimators and selecting a class of estimators, to computing costs and risks depending on unknown parameters.

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