Archive for foundations

The Seven Pillars of Statistical Wisdom [book review]

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , , , , on June 10, 2017 by xi'an

I remember quite well attending the ASA Presidential address of Stephen Stigler at JSM 2014, Boston, on the seven pillars of statistical wisdom. In connection with T.E. Lawrence’s 1926 book. Itself in connection with Proverbs IX:1. Unfortunately wrongly translated as seven pillars rather than seven sages.

As pointed out in the Acknowledgements section, the book came prior to the address by several years. I found it immensely enjoyable, first for putting the field in a (historical and) coherent perspective through those seven pillars, second for exposing new facts and curios about the history of statistics, third because of a literary style one would wish to see more often in scholarly texts and of a most pleasant design (and the list of reasons could go on for quite a while, one being the several references to Jorge Luis Borges!). But the main reason is to highlight the unified nature of Statistics and the reasons why it does not constitute a subfield of either Mathematics or Computer Science. In these days where centrifugal forces threaten to split the field into seven or more disciplines, the message is welcome and urgent.

Here are Stephen’s pillars (some comments being already there in the post I wrote after the address):

  1. aggregation, which leads to gain information by throwing away information, aka the sufficiency principle. One (of several) remarkable story in this section is the attempt by Francis Galton, never lacking in imagination, to visualise the average man or woman by superimposing the pictures of several people of a given group. In 1870!
  2. information accumulating at the √n rate, aka precision of statistical estimates, aka CLT confidence [quoting  de Moivre at the core of this discovery]. Another nice story is Newton’s wardenship of the English Mint, with musing about [his] potential exploiting this concentration to cheat the Mint and remain undetected!
  3. likelihood as the right calibration of the amount of information brought by a dataset [including Bayes’ essay as an answer to Hume and Laplace’s tests] and by Fisher in possible the most impressive single-handed advance in our field;
  4. intercomparison [i.e. scaling procedures from variability within the data, sample variation], from Student’s [a.k.a., Gosset‘s] t-test, better understood and advertised by Fisher than by the author, and eventually leading to the bootstrap;
  5. regression [linked with Darwin’s evolution of species, albeit paradoxically, as Darwin claimed to have faith in nothing but the irrelevant Rule of Three, a challenging consequence of this theory being an unobserved increase in trait variability across generations] exposed by Darwin’s cousin Galton [with a detailed and exhilarating entry on the quincunx!] as conditional expectation, hence as a true Bayesian tool, the Bayesian approach being more specifically addressed in (on?) this pillar;
  6. design of experiments [re-enters Fisher, with his revolutionary vision of changing all factors in Latin square designs], with an fascinating insert on the 18th Century French Loterie,  which by 1811, i.e., during the Napoleonic wars, provided 4% of the national budget!;
  7. residuals which again relate to Darwin, Laplace, but also Yule’s first multiple regression (in 1899), Fisher’s introduction of parametric models, and Pearson’s χ² test. Plus Nightingale’s diagrams that never cease to impress me.

The conclusion of the book revisits the seven pillars to ascertain the nature and potential need for an eight pillar.  It is somewhat pessimistic, at least my reading of it was, as it cannot (and presumably does not want to) produce any direction about this new pillar and hence about the capacity of the field of statistics to handle in-coming challenges and competition. With some amount of exaggeration (!) I do hope the analogy of the seven pillars that raises in me the image of the beautiful ruins of a Greek temple atop a Sicilian hill, in the setting sun, with little known about its original purpose, remains a mere analogy and does not extend to predict the future of the field! By its very nature, this wonderful book is about foundations of Statistics and therefore much more set in the past and on past advances than on the present, but those foundations need to move, grow, and be nurtured if the field is not to become a field of ruins, a methodology of the past!

La déraisonnable efficacité des mathématiques

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , on May 11, 2017 by xi'an

Although it went completely out of my mind, thanks to a rather heavy travel schedule, I gave last week a short interview about the notion of mathematical models, which got broadcast this week on France Culture, one of the French public radio channels. Within the daily La Méthode Scientifique show, which is a one-hour emission on scientific issues, always a [rare] pleasure to listen to. (Including the day they invited Claire Voisin.) The theme of the show that day was about the unreasonable effectiveness of mathematics, with the [classical] questioning of whether it is an efficient tool towards solving scientific (and inference?) problems because the mathematical objects pre-existed their use or we are (pre-)conditioned to use mathematics to solve problems. I somewhat sounded like a dog in a game of skittles, but it was interesting to listen to the philosopher discussing my relativistic perspective [provided you understand French!]. And I appreciated very much the way Céline Loozen the journalist who interviewed me sorted the chaff from the wheat in the original interview to make me sound mostly coherent! (A coincidence: Jean-Michel Marin got interviewed this morning on France Inter, the major public radio, about the Grothendieck papers.)

machine learning and the future of realism

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , on May 4, 2017 by xi'an

Giles and Cliff Hooker arXived a paper last week with this intriguing title. (Giles Hooker is an associate professor of statistics and biology at Cornell U, with an interesting blog on the notion of models, while Cliff Hooker is a professor of philosophy at Newcastle U, Australia.)

“Our conclusion is that simplicity is too complex”

The debate in this short paper is whether or not machine learning relates to a model. Or is it concerned with sheer (“naked”) prediction? And then does it pertain to science any longer?! While it sounds obvious at first, defining why science is more than prediction of effects given causes is much less obvious, although prediction sounds more pragmatic and engineer-like than scientific. (Furthermore, prediction has a somewhat negative flavour in French, being used as a synonym to divination and opposed to prévision.) In more philosophical terms, prediction offers no ontological feature. As for a machine learning structure like a neural network being scientific or a-scientific, its black box nature makes it much more the later than the former, in that it brings no explanation for the connection between input and output, between regressed and regressors. It further lacks the potential for universality of scientific models. For instance, as mentioned in the paper, Newton’s law of gravitation applies to any pair of weighted bodies, while a neural network built on a series of observations could not be assessed or guaranteed outside the domain where those observations are taken. Plus, would miss the simple square law established by Newton. Most fascinating questions, undoubtedly! Putting the stress on models from a totally different perspective from last week at the RSS.

As for machine learning being a challenge to realism, I am none the wiser after reading the paper. Utilising machine learning tools to produce predictions of causes given effects does not seem to modify the structure of the World and very little our understanding of it, since they do not bring explanation per se. What would lead to anti-realism is the adoption of those tools as substitutes for scientific theories and models.

principles or unprincipled?!

Posted in Books, Kids, pictures, Statistics, Travel with tags , , , , , , , on May 2, 2017 by xi'an

A lively and wide-ranging discussion during the Bayes, Fiducial, Frequentist conference was about whether or not we should look for principles. Someone mentioned Terry Speed (2016) claim that it does not help statistics in being principled. Against being efficient. Which gets quite close in my opinion to arguing in favour of a no-U-turn move to machine learning—which requires a significant amount of data to reach this efficiency, as Xiao-Li Meng mentioned—. The debate brought me back to my current running or droning argument on the need to accommodate [more] the difference between models and reality. Not throwing away statistics and models altogether, but developing assessments that are not fully chained to those models. While keeping probabilistic models to handle uncertainty. One pessimistic conclusion I drew from the discussion is that while we [as academic statisticians] may set principles and even teach our students how to run principled and ethical statistical analyses, there is not much we can do about the daily practice of users of statistics…

Fourth Bayesian, Fiducial, and Frequentist Conference

Posted in Books, pictures, Statistics, Travel, University life, Wines with tags , , , , , , , on March 29, 2017 by xi'an

Next May 1-3, I will attend the 4th Bayesian, Fiducial and Frequentist Conference at Harvard University (hopefully not under snow at that time of year), which is a meeting between philosophers and statisticians about foundational thinking in statistics and inference under uncertainty. This should be fun! (Registration is now open.)

Le bayésianisme aujourd’hui

Posted in Books, Statistics with tags , , , , , , , on September 19, 2016 by xi'an

A few years ago, I was asked by Isabelle Drouet to contribute a chapter to a multi-disciplinary book on the Bayesian paradigm, book that is now soon to appear. In French. It has this rather ugly title of Bayesianism today. Not that I had hear of Bayesianism or bayésianime previously. There are chapters on the Bayesian notion(s) of probability, game theory, statistics, on applications, and on the (potentially) Bayesian structure of human intelligence. Most of it is thus outside statistics, but I will certainly read through it when I receive my copy.

Validity and the foundations of statistical inference

Posted in Statistics with tags , , , , , , , , on July 29, 2016 by xi'an

Natesh pointed out to me this recent arXival with a somewhat grandiose abstract:

In this paper, we argue that the primary goal of the foundations of statistics is to provide data analysts with a set of guiding principles that are guaranteed to lead to valid statistical inference. This leads to two new questions: “what is valid statistical inference?” and “do existing methods achieve this?” Towards answering these questions, this paper makes three contributions. First, we express statistical inference as a process of converting observations into degrees of belief, and we give a clear mathematical definition of what it means for statistical inference to be valid. Second, we evaluate existing approaches Bayesian and frequentist approaches relative to this definition and conclude that, in general, these fail to provide valid statistical inference. This motivates a new way of thinking, and our third contribution is a demonstration that the inferential model framework meets the proposed criteria for valid and prior-free statistical inference, thereby solving perhaps the most important unsolved problem in statistics.

Since solving the “most important unsolved problem in statistics” sounds worth pursuing, I went and checked the paper‘s contents.

“To us, the primary goal of the foundations of statistics is to provide a set of guiding principles that, if followed, will guarantee validity of the resulting inference. Our motivation for writing this paper is to be clear about what is meant by valid inference and to provide the necessary principles to help data analysts achieve validity.”

Which can be interpreted in so many ways that it is somewhat meaningless…

“…if real subjective prior information is available, we recommend using it. However, there is an expanding collection of work (e.g., machine learning, etc) that takes the perspective that no real prior information is available. Even a large part of the literature claiming to be Bayesian has abandoned the interpretation of the prior as a serious part of the model, opting for “default” prior that “works.” Our choice to omit a prior from the model is not for the (misleading) purpose of being “objective”—subjectivity is necessary—but, rather, for the purpose of exploring what can be done in cases where a fully satisfactory prior is not available, to see what improvements can be made over the status quo.”

This is a pretty traditional criticism of the Bayesian approach, namely that if a “true” prior is provided (by whom?) then it is optimal to use it. But this amounts to turn the prior into another piece of the sampling distribution and is not in my opinion a Bayesian argument! Most of the criticisms in the paper are directed at objective Bayes approaches, with the surprising conclusion that, because there exist cases where no matching prior is available, “the objective Bayesian approach [cannot] be considered as a general framework for scientific inference.” (p.9)

Another section argues that a Bayesian modelling cannot describe a state of total ignorance. This is formally correct, which is why there is no such thing as a non-informative or the non-informative prior, as often discussed here, but is this truly relevant, in that the inference problem contains one way or another information about the parameter, for instance through a loss function or a pseudo-likelihood.

“This is a desirable property that most existing methods lack.”

The proposal central to the paper thesis is to replace posterior probabilities by belief functions b(.|X), called statistical inference, that are interpreted as measures of evidence about subsets A of the parameter space. If not necessarily as probabilities. This is not very novel, witness the works of Dempster, Shafer and subsequent researchers. And not very much used outside Bayesian and fiducial statistics because of the mostly impossible task of defining a function over all subsets of the parameter space. Because of the subjectivity of such “beliefs”, they will be “valid” only if they are well-calibrated in the sense of b(A|X) being sub-uniform, that is, more concentrated near zero than a uniform variate (i.e., small) under the alternative, i.e. when θ is not in A. At this stage, since this is a mix of a minimax and proper coverage condition, my interest started to quickly wane… Especially because the sub-uniformity condition is highly demanding, if leading to controls over the Type I error and the frequentist coverage. As often, I wonder at the meaning of a calibration property obtained over all realisations of the random variable and all values of the parameter. So for me stability is neither “desirable” nor “essential”. Overall, I have increasing difficulties in perceiving proper coverage as a relevant property. Which has no stronger or weaker meaning that the coverage derived from a Bayesian construction.

“…frequentism does not provide any guidance for selecting a particular rule or procedure.”

I agree with this assessment, which means that there is no such thing as frequentist inference, but rather a philosophy for assessing procedures. That the Gleser-Hwang paradox invalidates this philosophy sounds a bit excessive, however. Especially when the bounded nature of Bayesian credible intervals is also analysed as a failure. A more relevant criticism is the lack of directives for picking procedures.

“…we are the first to recognize that the belief function’s properties are necessary in order for the inferential output to satisfy the required validity property”

The construction of the “inferential model” proposed by the authors offers similarities withn fiducial inference, in that it builds upon the representation of the observable X as X=a(θ,U). With further constraints on the function a() to ensure the validity condition holds… An interesting point is that the functional connection X=a(θ,U) means that the nature of U changes once X is observed, albeit in a delicate manner outside a Bayesian framework. When illustrated on the Gleser-Hwang paradox, the resolution proceeds from an arbitrary choice of a one-dimensional summary, though. (As I am reading the paper, I realise it builds on other and earlier papers by the authors, papers that I cannot read for lack of time. I must have listned to a talk by one of the authors last year at JSM as this rings a bell. Somewhat.) In conclusion of a quick Sunday afternoon read, I am not convinced by the arguments in the paper and even less by the impression of a remaining arbitrariness in setting the resulting procedure.