Archive for The Bayesian Choice

how individualistic should statistics be?

Posted in Books, pictures, Statistics with tags , , , , , , , , , , , on November 5, 2015 by xi'an

keep-stats-subjectiveKeli Liu and Xiao-Li Meng completed a paper on the very nature of inference, to appear in The Annual Review of Statistics and Its Application. This paper or chapter is addressing a fundamental (and foundational) question on drawing inference based a sample on a new observation. That is, in making prediction. To what extent should the characteristics of the sample used for that prediction resemble those of the future observation? In his 1921 book, A Treatise on Probability, Keynes thought this similarity (or individualisation) should be pushed to its extreme, which led him to somewhat conclude on the impossibility of statistics and never to return to the field again. Certainly missing the incoming possibility of comparing models and selecting variables. And not building so much on the “all models are wrong” tenet. On the contrary, classical statistics use the entire data available and the associated model to run the prediction, including Bayesian statistics, although it is less clear how to distinguish between data and control there. Liu & Meng debate about the possibility of creating controls from the data alone. Or “alone” as the model behind always plays a capital role.

“Bayes and Frequentism are two ends of the same spectrum—a spectrum defined in terms of relevance and robustness. The nominal contrast between them (…) is a red herring.”

viemortrerbThe paper makes for an exhilarating if definitely challenging read. With a highly witty writing style. If only because the perspective is unusual, to say the least!, and requires constant mental contortions to frame the assertions into more traditional terms.  For instance, I first thought that Bayesian procedures were in agreement with the ultimate conditioning approach, since it conditions on the observables and nothing else (except for the model!). Upon reflection, I am not so convinced that there is such a difference with the frequentist approach in the (specific) sense that they both take advantage of the entire dataset. Either from the predictive or from the plug-in distribution. It all boils down to how one defines “control”.

“Probability and randomness, so tightly yoked in our minds, are in fact distinct concepts (…) at the end of the day, probability is essentially a tool for bookkeeping, just like the abacus.”

Some sentences from the paper made me think of ABC, even though I am not trying to bring everything back to ABC!, as drawing controls is the nature of the ABC game. ABC draws samples or control from the prior predictive and only keeps those for which the relevant aspects (or the summary statistics) agree with those of the observed data. Which opens similar questions about the validity and precision of the resulting inference, as well as the loss of information due to the projection over the summary statistics. While ABC is not mentioned in the paper, it can be used as a benchmark to walk through it.

“In the words of Jack Kiefer, we need to distinguish those problems with `luck data’ from those with `unlucky data’.”

keep-calm-and-condi-tionI liked very much recalling discussions we had with George Casella and Costas Goutis in Cornell about frequentist conditional inference, with the memory of Jack Kiefer still lingering around. However, I am not so excited about the processing of models here since, from what I understand in the paper (!), the probabilistic model behind the statistical analysis must be used to some extent in producing the control case and thus cannot be truly assessed with a critical eye. For instance, of which use is the mean square error when the model behind is unable to produce the observed data? In particular, the variability of this mean squared error is directly driven by this model. Similarly the notion of ancillaries is completely model-dependent. In the classification diagrams opposing robustness to relevance, all methods included therein are parametric. While non-parametric types of inference could provide a reference or a calibration ruler, at the very least.

Also, by continuously and maybe a wee bit heavily referring to the doctor-and-patient analogy, the paper is somewhat confusing as to which parts are analogy and which parts are methodology and to which type of statistical problem is covered by the discussion (sometimes it feels like all problems and sometimes like medical trials).

“The need to deliver individualized assessments of uncertainty are more pressing than ever.”

 A final question leads us to an infinite regress: if the statistician needs to turn to individualized inference, at which level of individuality should the statistician be assessed? And who is going to provide the controls then? In any case, this challenging paper is definitely worth reading by (only mature?) statisticians to ponder about the nature of the game!

Think Bayes: Bayesian Statistics Made Simple

Posted in Books, Kids, R, Statistics, University life with tags , , , , , , , , on October 27, 2015 by xi'an

Almost Bayes can!By some piece of luck, I came upon the book Think Bayes: Bayesian Statistics Made Simple, written by Allen B. Downey and published by Green Tea Press [which I could relate to No Starch Press, focussing on coffee!, which published Statistics Done Wrong that I reviewed a while ago] which usually publishes programming books with fun covers. The book is available on-line for free in pdf and html formats, and I went through it during a particularly exciting administrative meeting…

“Most books on Bayesian statistics use mathematical notation and present ideas in terms of mathematical concepts like calculus. This book uses Python code instead of math, and discrete approximations instead of continuous mathematics. As a result, what would be an integral in a math book becomes a summation, and most operations on probability distributions are simple loops.”

The book is most appropriately published in this collection as most of it concentrates on Python programming, with hardly any maths formula. In some sense similar to Jim Albert’s R book. Obviously, coming from maths, and having never programmed in Python, I find the approach puzzling, But just as obviously, I am aware—both from the comments on my books and from my experience on X validated—that a large group (majority?) of newcomers to the Bayesian realm find the mathematical approach to the topic a major hindrance. Hence I am quite open to this editorial choice as it is bound to include more people to think Bayes, or to think they can think Bayes.

“…in fewer than 200 pages we have made it from the basics of probability to the research frontier. I’m very happy about that.”

The choice made of operating almost exclusively through motivating examples is rather traditional in US textbooks. See e.g. Albert’s book. While it goes against my French inclination to start from theory and concepts and end up with illustrations, I can see how it operates in a programming book. But as always I fear it makes generalisations uncertain and understanding more shaky… The examples are per force simple and far from realistic statistics issues. Hence illustrates more the use of Bayesian thinking for decision making than for data analysis. To wit, those examples are about the Monty Hall problem and other TV games, some urn, dice, and coin models, blood testing, sport predictions, subway waiting times, height variability between men and women, SAT scores, cancer causality, a Geiger counter hierarchical model inspired by Jaynes, …, the exception being the final Belly Button Biodiversity dataset in the final chapter, dealing with the (exciting) unseen species problem in an equally exciting way. This may explain why the book does not cover MCMC algorithms. And why ABC is covered through a rather artificial normal example. Which also hides some of the maths computations under the carpet.

“The underlying idea of ABC is that two datasets are alike if they yield the same summary statistics. But in some cases, like the example in this chapter, it is not obvious which summary statistics to choose.¨

In conclusion, this is a very original introduction to Bayesian analysis, which I welcome for the reasons above. Of course, it is only an introduction, which should be followed by a deeper entry into the topic, and with [more] maths. In order to handle more realistic models and datasets.

model selection and multiple testing

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , on October 23, 2015 by xi'an

Ritabrata Dutta, Malgorzata Bogdan and Jayanta Ghosh recently arXived a survey paper on model selection and multiple testing. Which provides a good opportunity to reflect upon traditional Bayesian approaches to model choice. And potential alternatives. On my way back from Madrid, where I got a bit distracted when flying over the South-West French coast, from Biarritz to Bordeaux. Spotting the lake of Hourtain, where I spent my military training month, 29 years ago!

“On the basis of comparison of AIC and BIC, we suggest tentatively that model selection rules should be used for the purpose for which they were introduced. If they are used for other problems, a fresh justification is desirable. In one case, justification may take the form of a consistency theorem, in the other some sort of oracle inequality. Both may be hard to prove. Then one should have substantial numerical assessment over many different examples.”

The authors quickly replace the Bayes factor with BIC, because it is typically consistent. In the comparison between AIC and BIC they mention the connundrum of defining a prior on a nested model from the prior on the nesting model, a problem that has not been properly solved in my opinion. The above quote with its call to a large simulation study reminded me of the paper by Arnold & Loeppky about running such studies through ecdfs. That I did not see as solving the issue. The authors also discuss DIC and Lasso, without making much of a connection between those, or with the above. And then reach the parametric empirical Bayes approach to model selection exemplified by Ed George’s and Don Foster’s 2000 paper. Which achieves asymptotic optimality for posterior prediction loss (p.9). And which unifies a wide range of model selection approaches.

A second part of the survey considers the large p setting, where BIC is not a good approximation to the Bayes factor (when testing whether or not all mean entries are zero). And recalls that there are priors ensuring consistency for the Bayes factor in this very [restrictive] case. Then, in Section 4, the authors move to what they call “cross-validatory Bayes factors”, also known as partial Bayes factors and pseudo-Bayes factors, where the data is split to (a) make the improper prior proper and (b) run the comparison or test on the remaining data. They also show the surprising result that, provided the fraction of the data used to proper-ise the prior does not converge to one, the X validated Bayes factor remains consistent [for the special case above]. The last part of the paper concentrates on multiple testing but is more tentative and conjecturing about convergence results, centring on the differences between full Bayes and empirical Bayes. Then the plane landed in Paris and I stopped my reading, not feeling differently about the topic than when the plane started from Madrid.

seminar im München, am Max-Planck-Institut für Astrophysik

Posted in Statistics, Travel, University life with tags , , , , , , , , , , , , on October 15, 2015 by xi'an

On Friday, I give a talk in München on ABC model choice. At the Max-Planck Institute for Astrophysics. As coincidence go, I happen to talk the week after John Skilling gave a seminar there. On Bayesian tomography, not on nested sampling. And the conference organisers put the cover of the book Think Bayes: Bayesian Statistics Made Simple, written by Allen Downey, a book I reviewed yesterday night for CHANCE (soon to appear on the ‘Og!) [not that I understand the connection with the Max-Planck Institute or with my talk!, warum nicht?!] The slides are the same as in Oxford for SPA 2015:

Mathematical underpinnings of Analytics (theory and applications)

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , , , on September 25, 2015 by xi'an

“Today, a week or two spent reading Jaynes’ book can be a life-changing experience.” (p.8)

I received this book by Peter Grindrod, Mathematical underpinnings of Analytics (theory and applications), from Oxford University Press, quite a while ago. (Not that long ago since the book got published in 2015.) As a book for review for CHANCE. And let it sit on my desk and in my travel bag for the same while as it was unclear to me that it was connected with Statistics and CHANCE. What is [are?!] analytics?! I did not find much of a definition of analytics when I at last opened the book, and even less mentions of statistics or machine-learning, but Wikipedia told me the following:

“Analytics is a multidimensional discipline. There is extensive use of mathematics and statistics, the use of descriptive techniques and predictive models to gain valuable knowledge from data—data analysis. The insights from data are used to recommend action or to guide decision making rooted in business context. Thus, analytics is not so much concerned with individual analyses or analysis steps, but with the entire methodology.”

Barring the absurdity of speaking of a “multidimensional discipline” [and even worse of linking with the mathematical notion of dimension!], this tells me analytics is a mix of data analysis and decision making. Hence relying on (some) statistics. Fine.

“Perhaps in ten years, time, the mathematics of behavioural analytics will be common place: every mathematics department will be doing some of it.”(p.10)

First, and to start with some positive words (!), a book that quotes both Friedrich Nietzsche and Patti Smith cannot get everything wrong! (Of course, including a most likely apocryphal quote from the now late Yogi Berra does not partake from this category!) Second, from a general perspective, I feel the book meanders its way through chapters towards a higher level of statistical consciousness, from graphs to clustering, to hidden Markov models, without precisely mentioning statistics or statistical model, while insisting very much upon Bayesian procedures and Bayesian thinking. Overall, I can relate to most items mentioned in Peter Grindrod’s book, but mostly by first reconstructing the notions behind. While I personally appreciate the distanced and often ironic tone of the book, reflecting upon the author’s experience in retail modelling, I am thus wondering at which audience Mathematical underpinnings of Analytics aims, for a practitioner would have a hard time jumping the gap between the concepts exposed therein and one’s practice, while a theoretician would require more formal and deeper entries on the topics broached by the book. I just doubt this entry will be enough to lead maths departments to adopt behavioural analytics as part of their curriculum… Continue reading

Bayesian propaganda?

Posted in Books, Kids, pictures, Statistics, University life with tags , , , , , , , , , on April 20, 2015 by xi'an

“The question is about frequentist approach. Bayesian is admissable [sic] only by wrong definition as it starts with the assumption that the prior is the correct pre-information. James-Stein beats OLS without assumptions. If there is an admissable [sic] frequentist estimator then it will correspond to a true objective prior.”

I had a wee bit of a (minor, very minor!) communication problem on X validated, about a question on the existence of admissible estimators of the linear regression coefficient in multiple dimensions, under squared error loss. When I first replied that all Bayes estimators with finite risk were de facto admissible, I got the above reply, which clearly misses the point, and as I had edited the OP question to include more tags, the edited version was reverted with a comment about Bayesian propaganda! This is rather funny, if not hilarious, as (a) Bayes estimators are indeed admissible in the classical or frequentist sense—I actually fail to see a definition of admissibility in the Bayesian sense—and (b) the complete class theorems of Wald, Stein, and others (like Jack Kiefer, Larry Brown, and Jim Berger) come from the frequentist quest for best estimator(s). To make my point clearer, I also reproduced in my answer the Stein’s necessary and sufficient condition for admissibility from my book but it did not help, as the theorem was “too complex for [the OP] to understand”, which shows in fine the point of reading textbooks!

comparison of Bayesian predictive methods for model selection

Posted in Books, Statistics, University life with tags , , , , , , , , , on April 9, 2015 by xi'an

“Dupuis and Robert (2003) proposed choosing the simplest model with enough explanatory power, for example 90%, but did not discuss the effect of this threshold for the predictive performance of the selected models. We note that, in general, the relative explanatory power is an unreliable indicator of the predictive performance of the submodel,”

Juho Piironen and Aki Vehtari arXived a survey on Bayesian model selection methods that is a sequel to the extensive survey of Vehtari and Ojanen (2012). Because most of the methods described in this survey stem from Kullback-Leibler proximity calculations, it includes some description of our posterior projection method with Costas Goutis and Jérôme Dupuis. We indeed did not consider prediction in our papers and even failed to include consistency result, as I was pointed out by my discussant in a model choice meeting in Cagliari, in … 1999! Still, I remain fond of the notion of defining a prior on the embedding model and of deducing priors on the parameters of the submodels by Kullback-Leibler projections. It obviously relies on the notion that the embedding model is “true” and that the submodels are only approximations. In the simulation experiments included in this survey, the projection method “performs best in terms of the predictive ability” (p.15) and “is much less vulnerable to the selection induced bias” (p.16).

Reading the other parts of the survey, I also came to the perspective that model averaging makes much more sense than model choice in predictive terms. Sounds obvious stated that way but it took me a while to come to this conclusion. Now, with our mixture representation, model averaging also comes as a natural consequence of the modelling, a point presumably not stressed enough in the current version of the paper. On the other hand, the MAP model now strikes me as artificial and linked to a very rudimentary loss function. A loss that does not account for the final purpose(s) of the model. And does not connect to the “all models are wrong” theorem.


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