Archive for CHANCE

understanding elections through statistics [book review]

Posted in Books, Kids, R, Statistics, Travel with tags , , , , , , , , , , , , , , , , , , , , , , , , on October 12, 2020 by xi'an

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.]

the biggest bluff [not a book review]

Posted in Books with tags , , , , , , , , , , , on August 14, 2020 by xi'an

It came as a surprise to me that the book reviewed in the book review section of Nature of 25 June was a personal account of a professional poker player, The Biggest Bluff by Maria Konnikova.  (Surprise enough to write a blog entry!) As I see very little scientific impetus in studying the psychology of poker players and the associated decision making. Obviously, this is not a book review, but a review of the book review. (Although the NYT published a rather extensive extract of the book, from which I cannot detect anything deep from a game-theory viewpoint. Apart from the maybe-not-so-deep message that psychology matters a lot in poker…) Which does not bring much incentive for those uninterested (or worse) in money games like poker. Even when “a heap of Bayesian model-building [is] thrown in”, as the review mixes randomness and luck, while seeing the book as teaching the reader “how to play the game of life”, a type of self-improvement vending line one hardly expects to read in a scientific journal. (But again I have never understood the point in playing poker…)

Monte Carlo Markov chains

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , , , , , , , , , , on May 12, 2020 by xi'an

Darren Wraith pointed out this (currently free access) Springer book by Massimiliano Bonamente [whose family name means good spirit in Italian] to me for its use of the unusual Monte Carlo Markov chain rendering of MCMC.  (Google Trend seems to restrict its use to California!) This is a graduate text for physicists, but one could nonetheless expect more rigour in the processing of the topics. Particularly of the Bayesian topics. Here is a pot-pourri of memorable quotes:

“Two major avenues are available for the assignment of probabilities. One is based on the repetition of the experiments a large number of times under the same conditions, and goes under the name of the frequentist or classical method. The other is based on a more theoretical knowledge of the experiment, but without the experimental requirement, and is referred to as the Bayesian approach.”

“The Bayesian probability is assigned based on a quantitative understanding of the nature of the experiment, and in accord with the Kolmogorov axioms. It is sometimes referred to as empirical probability, in recognition of the fact that sometimes the probability of an event is assigned based upon a practical knowledge of the experiment, although without the classical requirement of repeating the experiment for a large number of times. This method is named after the Rev. Thomas Bayes, who pioneered the development of the theory of probability.”

“The likelihood P(B/A) represents the probability of making the measurement B given that the model A is a correct description of the experiment.”

“…a uniform distribution is normally the logical assumption in the absence of other information.”

“The Gaussian distribution can be considered as a special case of the binomial, when the number of tries is sufficiently large.”

“This clearly does not mean that the Poisson distribution has no variance—in that case, it would not be a random variable!”

“The method of moments therefore returns unbiased estimates for the mean and variance of every distribution in the case of a large number of measurements.”

“The great advantage of the Gibbs sampler is the fact that the acceptance is 100 %, since there is no rejection of candidates for the Markov chain, unlike the case of the Metropolis–Hastings algorithm.”

Let me then point out (or just whine about!) the book using “statistical independence” for plain independence, the use of / rather than Jeffreys’ | for conditioning (and sometimes forgetting \ in some LaTeX formulas), the confusion between events and random variables, esp. when computing the posterior distribution, between models and parameter values, the reliance on discrete probability for continuous settings, as in the Markov chain chapter, confusing density and probability, using Mendel’s pea data without mentioning the unlikely fit to the expected values (or, as put more subtly by Fisher (1936), “the data of most, if not all, of the experiments have been falsified so as to agree closely with Mendel’s expectations”), presenting Fisher’s and Anderson’s Iris data [a motive for rejection when George was JASA editor!] as a “a new classic experiment”, mentioning Pearson but not Lee for the data in the 1903 Biometrika paper “On the laws of inheritance in man” (and woman!), and not accounting for the discrete nature of this data in the linear regression chapter, the three page derivation of the Gaussian distribution from a Taylor expansion of the Binomial pmf obtained by differentiating in the integer argument, spending endless pages on deriving standard properties of classical distributions, this appalling mess of adding over the conditioning atoms with no normalisation in a Poisson experiment

P(X=4|\mu=0,1,2) = \sum_{\mu=0}^2 \frac{\mu^4}{4!}\exp\{-\mu\},

botching the proof of the CLT, which is treated before the Law of Large Numbers, restricting maximum likelihood estimation to the Gaussian and Poisson cases and muddling its meaning by discussing unbiasedness, confusing a drifted Poisson random variable with a drift on its parameter, as well as using the pmf of the Poisson to define an area under the curve (Fig. 5.2), sweeping the improperty of a constant prior under the carpet, defining a null hypothesis as a range of values for a summary statistic, no mention of Bayesian perspectives in the hypothesis testing, model comparison, and regression chapters, having one-dimensional case chapters followed by two-dimensional case chapters, reducing model comparison to the use of the Kolmogorov-Smirnov test, processing bootstrap and jackknife in the Monte Carlo chapter without a mention of importance sampling, stating recurrence results without assuming irreducibility, motivating MCMC by the intractability of the evidence, resorting to the term link to designate the current value of a Markov chain, incorporating the need for a prior distribution in a terrible description of the Metropolis-Hastings algorithm, including a discrete proof for its stationarity, spending many pages on early 1990’s MCMC convergence tests rather than discussing the adaptive scaling of proposal distributions, the inclusion of numerical tables [in a 2017 book] and turning Bayes (1763) into Bayes and Price (1763), or Student (1908) into Gosset (1908).

[Usual disclaimer about potential self-plagiarism: this post or an edited version of it could possibly appear later in my Books Review section in CHANCE. Unlikely, though!]

essentials of probability theory for statisticians

Posted in Books, Kids, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , on April 25, 2020 by xi'an

On yet another confined sunny lazy Sunday morning, I read through Proschan and Shaw’s Essentials of Probability Theory for Statisticians, a CRC Press book that was sent to me quite a while ago for review. The book was indeed published in 2016. Before moving to serious things, let me evacuate the customary issue with the cover. I have trouble getting the point of the “face on Mars” being adopted as the cover of a book on probability theory (rather than a book on, say, pareidolia). There is a brief paragraph on post-facto probability calculations, stating how meaningless the question of the probability of this shade appearing on a Viking Orbiter picture by “chance”, but this is so marginal I would have preferred any other figure from the book!

The book plans to cover the probability essentials for dealing with graduate level statistics and in particular convergence, conditioning, and paradoxes following from using non-rigorous approaches to probability. A range that completely fits my own prerequisite for statistics students in my classes and that of course involves the recourse to (Lebesgue) measure theory. And a goal that I find both commendable and comforting as my past experience with exchange students led me to the feeling that rigorous probability theory was mostly scrapped from graduate programs. While the book is not extremely formal, it provides a proper motivation for the essential need of measure theory to handle the complexities of statistical analysis and in particular of asymptotics. It thus relies as much as possible on examples that stem from or relate to statistics, even though most examples may appear as standard to senior readers. For instance the consistency of the sample median or a weak version of the Glivenko-Cantelli theorem. The final chapter is dedicated to applications (in the probabilist’ sense!) that emerged from statistical problems. I felt these final chapters were somewhat stretched compared with what they could have been, as for instance with the multiple motivations of the conditional expectation, but this simply makes for more material. If I had to teach this material to students, I would certainly rely on the book! in particular because of the repeated appearances of the quincunx for motivating non-Normal limites. (A typo near Fatou’s lemma missed the dominating measure. And I did not notice the Riemann notation dx being extended to the measure in a formal manner.)

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

a computational approach to statistical learning [book review]

Posted in Books, R, Statistics, University life with tags , , , , , , , , , , , , , , , , on April 15, 2020 by xi'an

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.]