Archive for CRC Press

a concise introduction to statistical inference [book review]

Posted in Statistics with tags , , , , , , , , , , on February 16, 2017 by xi'an

[Just to warn readers and avoid emails about Xi’an plagiarising Christian!, this book was sent to me by CRC Press for a review. To be published in CHANCE.]

This is an introduction to statistical inference. And with 180 pages, it indeed is concise! I could actually stop the review at this point as a concise review of a concise introduction to statistical inference, as I do not find much originality in this introduction, intended for “mathematically sophisticated first-time student of statistics”. Although sophistication is in the eye of the sophist, of course, as this book has margin symbols in the guise of integrals to warn of section using “differential or integral calculus” and a remark that the book is still accessible without calculus… (Integral calculus as in Riemann integrals, not Lebesgue integrals, mind you!) It even includes appendices with the Greek alphabet, summation notations, and exponential/logarithms.

“In statistics we often bypass the probability model altogether and simply specify the random variable directly. In fact, there is a result (that we won’t cover in detail) that tells us that, for any random variable, we can find an appropriate probability model.” (p.17)

Given its limited mathematical requirements, the book does not get very far in the probabilistic background of statistics methods, which makes the corresponding chapter not particularly helpful as opposed to a prerequisite on probability basics. Since not much can be proven without “all that complicated stuff about for any ε>0” (p.29). And makes defining correctly notions like the Central Limit Theorem impossible. For instance, Chebychev’s inequality comes within a list of admitted results. There is no major mistake in the chapter, even though mentioning that two correlated Normal variables are jointly Normal (p.27) is inexact.

“The power of a test is the probability that you do not reject a null that is in fact correct.” (p.120)

Most of the book follows the same pattern as other textbooks at that level, covering inference on a mean and a probability, confidence intervals, hypothesis testing, p-values, and linear regression. With some words of caution about the interpretation of p-values. (And the unfortunate inversion of the interpretation of power above.) Even mentioning the Cult [of Significance] I reviewed a while ago.

Given all that, the final chapter comes as a surprise, being about Bayesian inference! Which should make me rejoice, obviously, but I remain skeptical of introducing the concept to readers with so little mathematical background. And hence a very shaky understanding of a notion like conditional distributions. (Which reminds me of repeated occurrences on X validated when newcomers hope to bypass textbooks and courses to grasp the meaning of posteriors and such. Like when asking why Bayes Theorem does not apply for expectations.) I can feel the enthusiasm of the author for this perspective and it may diffuse to some readers, but apart from being aware of the approach, I wonder how much they carry away from this brief (decent) exposure. The chapter borrows from Lee (2012, 4th edition) and from Berger (1985) for the decision-theoretic part. The limitations of the exercise are shown for hypothesis testing (or comparison) by the need to restrict the parameter space to two possible values. And for decision making. Similarly, introducing improper priors and the likelihood principle [distinguished there from the law of likelihood] is likely to get over the head of most readers and clashes with the level of the previous chapters. (And I do not think this is the most efficient way to argue in favour of a Bayesian approach to the problem of statistical inference: I have now dropped all references to the likelihood principle from my lectures. Not because of the controversy, but simply because the students do not get it.) By the end of the chapter, it is unclear a neophyte would be able to spell out how one could specify a prior for one of the problems processed in the earlier chapters. The appendix on de Finetti’s formalism on personal probabilities is very much unlikely to help in this regard. While it sounds so far beyond the level of the remainder of the book.

inferential models: reasoning with uncertainty [book review]

Posted in Books, Statistics, University life with tags , , , , , , , , , on October 6, 2016 by xi'an

“the field of statistics (…) is still surprisingly underdeveloped (…) the subject lacks a solid theory for reasoning with uncertainty [and] there has been very little progress on the foundations of statistical inference” (p.xvi)

A book that starts with such massive assertions is certainly hoping to attract some degree of attention from the field and likely to induce strong reactions to this dismissal of the not inconsiderable amount of research dedicated so far to statistical inference and in particular to its foundations. Or even attarcting flak for not accounting (in this introduction) for the past work of major statisticians, like Fisher, Kiefer, Lindley, Cox, Berger, Efron, Fraser and many many others…. Judging from the references and the tone of this 254 pages book, it seems like the two authors, Ryan Martin and Chuanhai Liu, truly aim at single-handedly resetting the foundations of statistics to their own tune, which sounds like a new kind of fiducial inference augmented with calibrated belief functions. Be warned that five chapters of this book are built on as many papers written by the authors in the past three years. Which makes me question, if I may, the relevance of publishing a book on a brand-new approach to statistics without further backup from a wider community.

“…it is possible to calibrate our belief probabilities for a common interpretation by intelligent minds.” (p.14)

Chapter 1 contains a description of the new perspective in Section 1.4.2, which I find useful to detail here. When given an observation x from a Normal N(θ,1) model, the authors rewrite X as θ+Z, with Z~N(0,1), as in fiducial inference, and then want to find a “meaningful prediction of Z independently of X”. This seems difficult to accept given that, once X=x is observed, Z=X-θ⁰, θ⁰ being the true value of θ, which belies the independence assumption. The next step is to replace Z~N(0,1) by a random set S(Z) containing Z and to define a belief function bel() on the parameter space Θ by

bel(A|X) = P(X-S(Z)⊆A)

which induces a pseudo-measure on Θ derived from the distribution of an independent Z, since X is already observed. When Z~N(0,1), this distribution does not depend on θ⁰ the true value of θ… The next step is to choose the belief function towards a proper frequentist coverage, in the approximate sense that the probability that bel(A|X) be more than 1-α is less than α when the [arbitrary] parameter θ is not in A. And conversely. This property (satisfied when bel(A|X) is uniform) is called validity or exact inference by the authors: in my opinion, restricted frequentist calibration would certainly sound more adequate.

“When there is no prior information available, [the philosophical justifications for Bayesian analysis] are less than fully convincing.” (p.30)

“Is it logical that an improper “ignorance” prior turns into a proper “non-ignorance” prior when combined with some incomplete information on the whereabouts of θ?” (p.44)

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a mistake in a 1990 paper

Posted in Kids, Statistics, University life with tags , , , , , , , , on August 7, 2016 by xi'an

As we were working on the Handbook of mixture analysis with Sylvia Früwirth-Schnatter and Gilles Celeux today, near Saint-Germain des Près, I realised that there was a mistake in our 1990 mixture paper with Jean Diebolt [published in 1994], in that when we are proposing to use improper “Jeffreys” priors under the restriction that no component of the Gaussian mixture is “empty”, meaning that there are at least two observations generated from each component, the likelihood needs to be renormalised to be a density for the sample. This normalisation constant only depends on the weights of the mixture, which means that, when simulating from the full conditional distribution of the weights, there should be an extra-acceptance step to account for this correction. Of course, the term is essentially equal to one for a large enough sample but this remains a mistake nonetheless! It is funny that it remained undetected for so long in my most cited paper. Checking on Larry’s 1999 paper exploring the idea of excluding terms from the likelihood to allow for improper priors, I did not spot him using a correction either.

Measuring statistical evidence using relative belief [book review]

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , , , , , , on July 22, 2015 by xi'an

“It is necessary to be vigilant to ensure that attempts to be mathematically general do not lead us to introduce absurdities into discussions of inference.” (p.8)

This new book by Michael Evans (Toronto) summarises his views on statistical evidence (expanded in a large number of papers), which are a quite unique mix of Bayesian  principles and less-Bayesian methodologies. I am quite glad I could receive a version of the book before it was published by CRC Press, thanks to Rob Carver (and Keith O’Rourke for warning me about it). [Warning: this is a rather long review and post, so readers may chose to opt out now!]

“The Bayes factor does not behave appropriately as a measure of belief, but it does behave appropriately as a measure of evidence.” (p.87)

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Bayesian inference for partially identified models [book review]

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

“The crux of the situation is that we lack theoretical insight into even quite basic questions about what is going on. More particularly, we cannot sayy anything about the limiting posterior marginal distribution of α compared to the prior marginal distribution of α.” (p.142)

Bayesian inference for partially identified models is a recent CRC Press book by Paul Gustafson that I received for a review in CHANCE with keen interest! If only because the concept of unidentifiability has always puzzled me. And that I have never fully understood what I felt was a sort of joker card that a Bayesian model was the easy solution to the problem since the prior was compensating for the components of the parameter not identified by the data. As defended by Dennis Lindley that “unidentifiability causes no real difficulties in the Bayesian approach”. However, after reading the book, I am less excited in that I do not feel it answers this type of questions about non-identifiable models and that it is exclusively centred on the [undoubtedly long-term and multifaceted] research of the author on the topic.

“Without Bayes, the feeling is that all the data can do is locate the identification region, without conveying any sense that some values in the region are more plausible than others.” (p.47)

Overall, the book is pleasant to read, with a light and witty style. The notational conventions are somewhat unconventional but well explained, to distinguish θ from θ* from θ. The format of the chapters is quite similar with a definition of the partially identified model, an exhibition of the transparent reparameterisation, the computation of the limiting posterior distribution [of the non-identified part], a demonstration [which it took me several iterations as the English exhibition rather than the French proof, pardon my French!]. Chapter titles suffer from an excess of the “further” denomination… The models themselves are mostly of one kind, namely binary observables and non-observables leading to partially observed multinomials with some non-identifiable probabilities. As in missing-at-random models (Chapter 3). In my opinion, it is only in the final chapters that the important questions are spelled-out, not always faced with a definitive answer. In essence, I did not get from the book (i) a characterisation of the non-identifiable parts of a model, of the  identifiability of unidentifiability, and of the universality of the transparent reparameterisation, (ii) a tool to assess the impact of a particular prior and possibly to set it aside, and (iii) a limitation to the amount of unidentifiability still allowing for coherent inference. Hence, when closing the book, I still remain in the dark (or at least in the grey) on how to handle partially identified models. The author convincingly argues that there is no special advantage to using a misspecified if identifiable model to a partially identified model, for this imbues false confidence (p.162), however we also need the toolbox to verify this is indeed the case.

“Given the data we can turn the Bayesian computational crank nonetheless and see what comes out.” (p.xix)

“It is this author’s contention that computation with partially identified models is a “bottleneck” issue.” (p.141)

Bayesian inference for partially identified models is particularly concerned about computational issues and rightly so. It is however unclear to me (without more time to invest investigating the topic) why the “use of general-purpose software is limited to the [original] parametrisation” (p.24) and why importance sampling would do better than MCMC on a general basis. I would definitely have liked more details on this aspect. There is a computational considerations section at the end of the book, but it remains too allusive for my taste. My naïve intuition would be that the lack of identifiability leads to flatter posterior and hence to easier MCMC moves, but Paul Gustafson reports instead bad mixing from standard MCMC schemes (like WinBUGS).

In conclusion, the book opens a new perspective on the relevance of partially identifiable models, trying to lift the stigma associated with them, and calls for further theory and methodology to deal with those. Here are the author’s final points (p.162):

  • “Identification is nuanced. Its absence does not preclude a parameter being well estimated, not its presence guarantee a parameter can be well estimated.”
  • “If we really took limitations of study designs and data quality seriously, then partially identifiable models would crop up all the time in a variety of scientific fields.”
  • “Making modeling assumptions for the sole purpose of gaining full identification can be a mug’s game (…)”
  • “If we accept partial identifiability, then consequently we need to regard sample size differently. There are profound implications of posterior variance tending to a positive limit as the sample size grows.”

These points may be challenging enough to undertake to read Bayesian inference for partially identified models in order to make one’s mind about their eventual relevance in statistical modelling.

[Disclaimer about potential self-plagiarism: this post will also be published as a book review in my CHANCE column. ]

Statistical analyses using R

Posted in Books, R, Statistics with tags , , , , , on April 22, 2011 by xi'an

Another book I received from the Short Book Reviews section of the International Statistical Review is Everitt’s and Hothorn’s Handbook of statistical analyses using R. Here is a [blog-ified] version of my book review.

This book is the second (blue) edition of a successful (violet) handbook that can benefit a wide audience interested in using R for its data analysis. (After I wrote the review, I saw this appropriate analysis of the first edition.) It covers most of non-Bayesian statistical methods, with forays into exploratory data analysis with tools like principal components, clustering and bagging/boosting. As reflected in the long list of chapters, the coverage is quite extensive and only missing specialised statistical domains like time-series (apart from longitudinal data), econometrics (except for generalised linear models), and signal processing. Beside the absence of a Bayesian perspective (only mentioned in connection with BIC and the mclust package, while the Bayesian formalism would be a natural tool for analysing mixed models), I miss some material on simulation, the only entry found in the book being bootstrap (pages 153-154).

Given its title and emphasis on analyses, the book is logically associated with an R package HSAUR2 [if there is an intended pun, I missed it!] and works according to a fixed pattern: each chapter (1) starts with a description of a few datasets, (2) summarises the statistical main issues in one or two pages, and then (3) engages into an R analysis. As the complexity increases with the chapter number, the authors are relying more and more on specialised packages that need to be downloaded by the reader. I have no objection with this pedagogical choice, especially when considering that the packages are mostly recent. I would however have like a bit more details about those packages or at least about their main function, as the reader is left to experiment solely from the line of code provided in the handbook. (In contrast, a few passages are a bit “geeky” and require a deeper understanding of R objects than casual readers master. Also, using layout instead of par(mfrow=… is not that obvious.) My only criticism of the book at this level is the puzzling insistence on including all the datasets used therein in the form of tables. I frankly fail to see the point in spending so many pages on those tables given that they all are available from the HSAUR2 package. A page of further explanation, of background or of statistical theory would have been much more beneficial to any reader, in my opinion! The same criticism applies to the few exercises found at the end of each chapter. (The most glaring use of a table occurs in the graphical display chapter, of course! The authors rely on a dataset about the 50 north-American States and list the data instead of illustrating the use of a map….)

In conclusion, I find the book by Everitt and Hothorn quite pleasant and bound to fit its purpose. The layout and presentation is nice (with a single noticeable mishap on page 332 caused by Darwin’s tree of life.) It should appeal to all readers as it contains a wealth of information about the use of R for statistical analysis. Included seasoned R users: When reading the first chapters, I found myself scribbling small light-bulbs in the margin to point out features of R I was not aware of. (In particular, the authors mentioned the option type=”n” for plot that R-bloggers signalled as the most useful option for plotting.) In addition, the book is quite handy for a crash introduction to statistics for (well-enough motivated) non-statisticians. (This post has also appeared on Statistical Forum on April 20.)