revisiting the Gelman-Rubin diagnostic

Just before Xmas, Dootika Vats (Warwick) and Christina Knudson arXived a paper on a re-evaluation of the ultra-popular 1992 Gelman and Rubin MCMC convergence diagnostic. Which compares within-variance and between-variance on parallel chains started from hopefully dispersed initial values. Or equivalently an under-estimating and an over-estimating estimate of the MCMC average. In this paper, the authors take advantage of the variance estimators developed by Galin Jones, James Flegal, Dootika Vats and co-authors, which are batch mean estimators consistently estimating the asymptotic variance. They also discuss the choice of a cut-off on the ratio R of variance estimates, i.e., how close to one need it be? By relating R to the effective sample size (for which we also have reservations), which gives another way of calibrating the cut-off. The main conclusion of the study is that the recommended 1.1 bound is too large for a reasonable proximity to the true value of the Bayes estimator (Disclaimer: The above ABCruise header is unrelated with the paper, apart from its use of the Titanic dataset!)

In fact, I have other difficulties than setting the cut-off point with the original scheme as a way to assess MCMC convergence or lack thereof, among which

1. its dependence on the parameterisation of the chain and on the estimation of a specific target function
2. its dependence on the starting distribution which makes the time to convergence not absolutely meaningful
3. the confusion between getting to stationarity and exploring the whole target
4. its missing the option to resort to subsampling schemes to attain pseudo-independence or scale time to convergence (albeit see 3. above)
5. a potential bias brought by the stopping rule.

the future of conferences

The last issue of Nature for 2018 offers a stunning collection of science photographs, ten portraits of people who mattered (for the editorial board of Nature), and a collection of journalists’ entries on scientific conferences. The later point leading to interesting questioning on the future of conferences, some of which relate to earlier entries on this blog. Like attempts to make them having a lesser carbon footprint, by only attending focused conferences and workshops, warning about predatory ones, creating local hives on different continents that can partake of all talks but reduce travel and size and still allow for exchanges person to person, multiply the meetings and opportunities around a major conference to induce “only” one major trip (as in the past summer of British conferences, or the incoming geographical combination of BNP and O’Bayes 2019), cut the traditional dreary succession of short talks in parallel in favour of “unconferences” where participants set communally the themes and  structure of the meeting (but ware the dangers of bias brought by language, culture, seniority!). Of course, this move towards new formats will meet opposition from several corners, including administrators who too often see conferences as a pretense for paid vacations and refuse supporting costs without a “concrete” proof of work in the form of a presentation.Another aspect of conference was discussed there, namely the art of delivering great talks. Which is indeed more an art than a science, since the impact will not only depend on the speaker and the slides, but also on the audience and the circumstances. As years pile on, I am getting less stressed and probably too relaxed about giving talks, but still rarely feel I have reached toward enough of the audience. And still falling too easily for the infodump mistake… Which reminds me of a recent column in Significance (although I cannot link to it!), complaining about “finding it hard or impossible to follow many presentations, particularly those that involved a large number of equations.” Which sounds strange to me as on the opposite I quickly loose track in talks with no equations. And as mathematical statistics or probability issues seems to imply the use of maths symbols and equations. (This reminded me of a short course I gave once in a undisclosed location, where a portion of the audience left after the first morning, due to my use of “too many Greek letters”.) Actually, I am always annoyed at apologies for using proper maths notations, since they are the tools of our trade.Another entry of importance in this issue of Nature is an interview with Katherine Heller and Hal Daumé, as first chairs for diversity and inclusion at N[eur]IPS. Where they discuss the actions taken since the previous NIPS 2017 meeting to address the lack of inclusiveness and the harassment cases exposed there, first by Kristian Lum, Lead Statistician at the Human Rights Data Analysis Group (HRDAG), whose blog denunciation set the wheels turning towards a safer and better environment (in stats as well as machine-learning). This included the [last minute] move towards renaming the conference as NeuroIPS to avoid sexual puns on the former acronym (which as a non-native speaker I missed until it was pointed out to me!). Judging from the feedback it seems that the wheels have indeed turned a significant amount and hopefully will continue its progress.

and it only gets worse…

““This is absolutely the stupidest thing ever,” said Antar Davis, 23, a former zookeeper who showed up in the elephant house on Friday to take one last look at Maharani, a 9,100-pound Asian elephant, before the zoo closed.” The New York Times, Dec 29, 2018

“The Trump administration has stopped cooperating with UN investigators over potential human rights violations occurring inside America [and] ceased to respond to official complaints from UN special rapporteurs, the network of independent experts who act as global watchdogs on fundamental issues such as poverty, migration, freedom of expression and justice.” The Guardian, Jan 4, 2019

“I know more about drones than anybody,” he said (…) Mr. Trump took the low number [of a 16% approval in Europe] as a measure of how well he is doing in the United States. “If I were popular in Europe, I wouldn’t be doing my job.”” The New York Times, Jan 3, 2019

““Any deaths of children or others at the border are strictly the fault of the Democrats and their pathetic immigration policies that allow people to make the long trek thinking they can enter our country illegally.” The New York Times, Dec 30, 2018

Ka [book review]

My last book of the year (2018), which I finished one hour before midnight, on 31 December! Ka is a book about a crow, or rather, a  Crow, Dar Oakley (or, in full, Dar of the Oak by the Lea), told from his viewpoint, and spanning all of Anthropocene, for Dar Oakley is immortal [sort of] and able to communicate with humans (and other birds, like Ravens. And coyotes). This summary of the plot may sound of limited appeal, but this may be the best book I read this past year. The Washington Post offers a critical entry into Ka that is much better than anything I can state about it. Not only it is about Crows and Ravens, fascinating social birds with a highly developed vocabulary that reflects the hierarchies in these avian societies. But it also offers another view on the doomed history of mankind, to which Crows seem irremediably linked and with whom  Dar Oakley is sharing more that a territory. As so acutely perceived in another review from Locus, the beauty of the book and the genius of the writer, John Crowley, is to translate an alien intelligence in terms intelligible to the reader.

“A crow alone is no crow.”

A fairly, faery, unique, strangely moving, book, thus, that cannot suffer to be labelled into a category like fantasy or poetry or philosophical tale. Reflecting on the solitude brought by knowledge and communicating with another race. And of the bittersweet pain brought by immortality that makes Dar Oakley seek a former mate in the kingdom of dead Crows. An imperfect, fallible character, a perfect messenger of Death to accompany humanity on its last steps.

prepaid ABC

Merijn Mestdagha, Stijn Verdoncka, Kristof Meersa, Tim Loossensa, and Francis Tuerlinckx from the KU Leuven, some of whom I met during a visit to its Wallon counterpart Louvain-La-Neuve, proposed and arXived a new likelihood-free approach based on saving simulations on a large scale for future users. Future users interested in the same model. The very same model. This makes the proposal quite puzzling as I have no idea as to when situations with exactly the same experimental conditions, up to the sample size, repeat over and over again. Or even just repeat once. (Some particular settings may accommodate for different sample sizes and the same prepaid database, but others as in genetics clearly do not.) I am sufficiently puzzled to suspect I have missed the message of the paper.

“In various fields, statistical models of interest are analytically intractable. As a result, statistical inference is greatly hampered by computational constraint s. However, given a model, different users with different data are likely to perform similar computations. Computations done by one user are potentially useful for other users with different data sets. We propose a pooling of resources across researchers to capitalize on this. More specifically, we preemptively chart out the entire space of possible model outcomes in a prepaid database. Using advanced interpolation techniques, any individual estimation problem can now be solved on the spot. The prepaid method can easily accommodate different priors as well as constraints on the parameters. We created prepaid databases for three challenging models and demonstrate how they can be distributed through an online parameter estimation service. Our method outperforms state-of-the-art estimation techniques in both speed (with a 23,000 to 100,000-fold speed up) and accuracy, and is able to handle previously quasi inestimable models.”

I foresee potential difficulties with this proposal, like compelling all future users to rely on the same summary statistics, on the same prior distributions (the “representative amount of parameter values”), and requiring a massive storage capacity. Plus furthermore relying at its early stage on the most rudimentary form of an ABC algorithm (although not acknowledged as such), namely the rejection one. When reading the description in the paper, the proposed method indeed selects the parameters (simulated from a prior or a grid) that are producing pseudo-observations that are closest to the actual observations (or their summaries s). The subsample thus constructed is used to derive a (local) non-parametric or machine-learning predictor s=f(θ). From which a point estimator is deduced by minimising in θ a deviance d(s⁰,f(θ)).

The paper does not expand much on the theoretical justifications of the approach (including the appendix that covers a formal situation where the prepaid grid conveniently covers the observed statistics). And thus does not explain on which basis confidence intervals should offer nominal coverage for the prepaid method. Instead, the paper runs comparisons with Simon Wood’s (2010) synthetic likelihood maximisation (Ricker model with three parameters), the rejection ABC algorithm (species dispersion trait model with four parameters), while the Leaky Competing Accumulator (with four parameters as well) seemingly enjoys no alternative. Which is strange since the first step of the prepaid algorithm is an ABC step, but I am unfamiliar with this model. Unsurprisingly, in all these cases, given that the simulation has been done prior to the computing time for the prepaid method and not for either synthetic likelihood or ABC, the former enjoys a massive advantage from the start.

“The prepaid method can be used for a very large number of observations, contrary to the synthetic likelihood or ABC methods. The use of very large simulated data sets allows investigation of large-sample properties of the estimator”

To return to the general proposal and my major reservation or misunderstanding, for different experiments, the (true or pseudo-true) value of the parameter will not be the same, I presume, and hence the region of interest [or grid] will differ. While, again, the computational gain is de facto obvious [since the costly production of the reference table is not repeated], and, to repeat myself, makes the comparison with methods that do require a massive number of simulations from scratch massively in favour of the prepaid option, I do not see a convenient way of recycling these prepaid simulations for another setting, that is, when some experimental factors, sample size or collection, or even just the priors, do differ. Again, I may be missing the point, especially in a specific context like repeated psychological experiments.

While this may have some applications in reproducibility (but maybe not, if the goal is in fact to detect cherry-picking), I see very little use in repeating the same statistical model on different datasets. Even repeating observations will require additional nuisance parameters and possibly perturb the likelihood and/or posterior to large extents.

statistics in Nature [a tale of the two Steves]

In the 29 November issue of Nature, Stephen Senn (formerly at Glasgow) wrote an article about the pitfalls of personalized medicine, for the statistics behind the reasoning are flawed.

“What I take issue with is the de facto assumption that the differential response to a drug is consistent for each individual, predictable and based on some stable property, such as a yet-to-be-discovered genetic variant.”S. Senn

One (striking) reason being that the studies rest on a sort of low-level determinism that does not account for many sources of variability. Over-confidence in causality results. Stephen argues that improvement lies in insisting on repeated experiments on the same subjects (with an increased challenge in modelling since this requires longitudinal models with dependent observations). And to “drop the use of dichotomies”, favouring instead continuous modeling of measurements.

And in the 6 December issue, Steven Goodman calls (in the World view tribune) for probability statements to be attached as confidence indices to scientific claims. That he takes great pain to distinguish from p-values and links with Bayesian analysis. (Bayesian analysis that Stephen regularly objects to.) While I applaud the call, I am quite pessimistic about the follow-up it will generate, the primary reply being that posterior probabilities can be manipulated as well as p-values. And that Bayesian probabilities are not “real” probabilities (dixit Don Fraser or Deborah Mayo).

Markov Chains [not a book review]

As Randal Douc and Éric Moulines are both very close friends and two authors of this book on Markov chains,  I cannot engage into a regular book review! Judging from the table of contents, the coverage is not too dissimilar to the now classic Markov chain Stochastic Stability book by Sean Meyn and the late Richard Tweedie (1994), called the Bible of Markov chains by Peter Glynn, with more emphasis on convergence matters and a more mathematical perspective. The 757 pages book also includes a massive appendix on maths and probability background. As indicated in the preface, “the reason [the authors] thought it would be useful to write a new book is to survey some of the developments made during the 25 years that have elapsed since the publication of Meyn and Tweedie (1993b).” Connecting with the theoretical developments brought by MCMC methods. Like subgeometric rates of convergence to stationarity, sample paths, limit theorems, and concentration inequalities. The book also reflects on the numerous contributions of the authors to the field. Hence a perfect candidate for teaching Markov chains to mathematically well-prepared. graduate audiences. Congrats to the authors!