Poisson-Belgium 0-0

“Statistical match predictions are more accurate than many people realize (…) For the upcoming Qatar World Cup, Penn’s model suggests that Belgium (…) has the highest chances of raising the famous trophy, followed by Brazil”

Even Nature had to get entries on the current football World cup, with a paper on data-analytics reaching football coaches and teams. This is not exactly prime news, as I remember visiting the Department of Statistics of the University of Glasgow in the mid 1990’s and chatting with a very friendly doctoral student who was consulting for the Glasgow Rangers (or Celtics?!) on the side at the time. And went back to Ireland to continue with a local team (Galway?!).

The paper reports on different modellings, including one double-Poisson model by (PhD) Matthew Penn from Oxford and (maths undergraduate) Joanna Marks from Warwick, which presumably resemble the double-Poisson version set by Leonardo Egidi et al. and posted on Andrews’ blog a few days ago. Following an earlier model by my friends Karlis & Ntzoufras in 2003. While predictive models can obviously fail, this attempt is missing Belgium, Germany, Switzerland, Mexico, Uruguay, and Denmark early elimination from the cup. One possible reason imho is that national teams do not play that often when players are employed by different clubs in many counties, hence are hard to assess, but I cannot claim any expertise or interest in the game.

exxon prediction [xkcd]

understanding elections through statistics [book review]

A book to read most urgently if hoping to take an informed decision by 03 November! Written by a political scientist cum statistician, Ole Forsberg. (If you were thinking of another political scientist cum statistician, he wrote red state blue state a while ago! And is currently forecasting the outcome of the November election for The Economist.)

“I believe [omitting educational level] was the main reason the [Brexit] polls were wrong.”

The first part of the book is about the statistical analysis of opinion polls (assuming their outcome is given, rather than designing them in the first place). And starting with the Scottish independence referendum of 2014. The first chapter covering the cartoon case of simple sampling from a population, with or without replacement, Bayes and non-Bayes. In somewhat too much detail imho given that this is an unrealistic description of poll outcomes. The second chapter expands to stratified sampling (with confusing title [Polling 399] and entry, since it discusses repeated polls that are not processed in said chapter). Mentioning the famous New York Times experiment where five groups of pollsters analysed the same data, making different decisions in adjusting the sample and identifying likely voters, and coming out with a range of five points in the percentage. Starting to get a wee bit more advanced when designing priors for the population proportions. But still studying a weighted average of the voting intentions for each category. Chapter three reaches the challenging task of combining polls, with a 2017 (South) Korea presidential election as an illustration, involving five polls. It includes a solution to handling older polls by proposing a simple linear regression against time. Chapter 4 sums up the challenges of real-life polling by examining the disastrous 2016 Brexit referendum in the UK. Exposing for instance the complicated biases resulting from polling by phone or on-line. The part that weights polling institutes according to quality does not provide any quantitative detail. (And also a weird averaging between the levels of “support for Brexit” and “maybe-support for Brexit”, see Fig. 4.5!) Concluding as quoted above that missing the educational stratification was the cause for missing the shock wave of referendum day is a possible explanation, but the massive difference in turnover between the age groups, itself possibly induced by the reassuring figures of the published polls and predictions, certainly played a role in missing the (terrible) outcome.

“The fabricated results conformed to Benford’s law on first digits, but failed to obey Benford’s law on second digits.” Wikipedia

The second part of this 200 page book is about election analysis, towards testing for fraud. Hence involving the ubiquitous Benford law. Although applied to the leading digit which I do not think should necessarily follow Benford law due to both the varying sizes and the non-uniform political inclinations of the voting districts (of which there are 39 for the 2009 presidential Afghan election illustration, although the book sticks at 34 (p.106)). My impression was that instead lesser digits should be tested. Chapter 4 actually supports the use of the generalised Benford distribution that accounts for differences in turnouts between the electoral districts. But it cannot come up with a real-life election where the B test points out a discrepancy (and hence a potential fraud). Concluding with the author’s doubt [repeated from his PhD thesis] that these Benford tests “are specious at best”, which makes me wonder why spending 20 pages on the topic. The following chapter thus considers other methods, checking for differential [i.e., not-at-random] invalidation by linear and generalised linear regression on the supporting rate in the district. Once again concluding at no evidence of such fraud when analysing the 2010 Côte d’Ivoire elections (that led to civil war). With an extension in Chapter 7 to an account for spatial correlation. The book concludes with an analysis of the Sri Lankan presidential elections between 1994 and 2019, with conclusions of significant differential invalidation in almost every election (even those not including Tamil provinces from the North).

R code is provided and discussed within the text. Some simple mathematical derivations are found, albeit with a huge dose of warnings (“math-heavy”, “harsh beauty”) and excuses (“feel free to skim”, “the math is entirely optional”). Often, one wonders at the relevance of said derivations for the intended audience and the overall purpose of the book. Nonetheless, it provides an interesting entry on (relatively simple) models applied to election data and could certainly be used as an original textbook on modelling aggregated count data, in particular as it should spark the interest of (some) students.

[Disclaimer about potential self-plagiarism: this post or an edited version will eventually appear in my Books Review section in CHANCE.]

focused Bayesian prediction

In this fourth session of our One World ABC Seminar, my friend and coauthor Gael Martin, gave an after-dinner talk on focused Bayesian prediction, more in the spirit of Bissiri et al. than following a traditional ABC approach.  because along with Ruben Loaiza-Maya and [my friend and coauthor] David Frazier, they consider the possibility of a (mild?) misspecification of the model. Using thus scoring rules à la Gneiting and Raftery. Gael had in fact presented an earlier version at our workshop in Oaxaca, in November 2018. As in other solutions of that kind, difficulty in weighting the score into a distribution. Although asymptotic irrelevance, direct impact on the current predictions, at least for the early dates in the time series… Further calibration of the set of interest A. Or the focus of the prediction. As a side note the talk perfectly fits the One World likelihood-free seminar as it does not use the likelihood function!

“The very premise of this paper is that, in reality, any choice of predictive class is such that the truth is not contained therein, at which point there is no reason to presume that the expectation of any particular scoring rule will be maximized at the truth or, indeed, maximized by the same predictive distribution that maximizes a different (expected) score.”

This approach requires the proxy class to be close enough to the true data generating model. Or in the word of the authors to be plausible predictive models. And to produce the true distribution via the score as it is proper. Or the closest to the true model in the misspecified family. I thus wonder at a possible extension with a non-parametric version, the prior being thus on functionals rather than parameters, if I understand properly the meaning of Π(P_θ). (Could the score function be misspecified itself?!) Since the score is replaced with its empirical version, the implementation is  resorting to off-the-shelf MCMC. (I wonder for a few seconds if the approach could be seen as a pseudo-marginal MCMC but the estimation is always based on the same observed sample, hence does not directly fit the pseudo-marginal MCMC framework.)

[Notice: Next talk in the series is tomorrow, 11:30am GMT+1.]

a computational approach to statistical learning [book review]

This book was sent to me by CRC Press for review for CHANCE. I read it over a few mornings while [confined] at home and found it much more computational than statistical. In the sense that the authors go quite thoroughly into the construction of standard learning procedures, including home-made R codes that obviously help in understanding the nitty-gritty of these procedures, what they call try and tell, but that the statistical meaning and uncertainty of these procedures remain barely touched by the book. This is not uncommon to the machine-learning literature where prediction error on the testing data often appears to be the final goal but this is not so traditionally statistical. The authors introduce their work as (a computational?) supplementary to Elements of Statistical Learning, although I would find it hard to either squeeze both books into one semester or dedicate two semesters on the topic, especially at the undergraduate level.

Each chapter includes an extended analysis of a specific dataset and this is an asset of the book. If sometimes over-reaching in selling the predictive power of the procedures. Printed extensive R scripts may prove tiresome in the long run, at least to me, but this may simply be a generational gap! And the learning models are mostly unidimensional, see eg the chapter on linear smoothers with imho a profusion of methods. (Could someone please explain the point of Figure 4.9 to me?) The chapter on neural networks has a fairly intuitive introduction that should reach fresh readers. Although meeting the handwritten digit data made me shift back to the late 1980’s, when my wife was working on automatic character recognition. But I found the visualisation of the learning weights for character classification hinting at their shape (p.254) most alluring!

Among the things I am missing when reading through this book, a life-line on the meaning of a statistical model beyond prediction, attention to misspecification, uncertainty and variability, especially when reaching outside the range of the learning data, and further especially when returning regression outputs with significance stars, discussions on the assessment tools like the distance used in the objective function (for instance lacking in scale invariance when adding errors on the regression coefficients) or the unprincipled multiplication of calibration parameters, some asymptotics, at least one remark on the information loss due to splitting the data into chunks, giving some (asymptotic) substance when using “consistent”, waiting for a single page 319 to see the “data quality issues” being mentioned. While the methodology is defended by algebraic and calculus arguments, there is very little on the probability side, which explains why the authors consider that the students need “be familiar  with the concepts of expectation, bias and variance”. And only that. A few paragraphs on the Bayesian approach are doing more harm than well, especially with so little background in probability and statistics.

The book possibly contains the most unusual introduction to the linear model I can remember reading: Coefficients as derivatives… Followed by a very detailed coverage of matrix inversion and singular value decomposition. (Would not sound like the #1 priority were I to give such a course.)

The inevitable typo “the the” was found on page 37! A less common typo was Jensen’s inequality spelled as “Jenson’s inequality”. Both in the text (p.157) and in the index, followed by a repetition of the same formula in (6.8) and (6.9). A “stwart” (p.179) that made me search a while for this unknown verb. Another typo in the Nadaraya-Watson kernel regression, when the bandwidth h suddenly turns into n (and I had to check twice because of my poor eyesight!). An unusual use of partition where the sets in the partition are called partitions themselves. Similarly, fluctuating use of dots for products in dimension one, including a form of ⊗ for matricial product (in equation (8.25)) followed next page by the notation for the Hadamard product. I also suspect the matrix K in (8.68) is missing 1’s or am missing the point, since K is the number of kernels on the next page, just after a picture of the Eiffel Tower…) A surprising number of references for an undergraduate textbook, with authors sometimes cited with full name and sometimes cited with last name. And technical reports that do not belong to this level of books. Let me add the pedant remark that Conan Doyle wrote more novels “that do not include his character Sherlock Holmes” than novels which do include Sherlock.

[Disclaimer about potential self-plagiarism: this post or an edited version will eventually appear in my Books Review section in CHANCE.]