statistical modeling with R [book review]

Statistical Modeling with R (A dual frequentist and Bayesian approach for life scientists) is a recent book written by Pablo Inchausti, from Uruguay. In a highly personal and congenial style (witness the preface), with references to (fiction) books that enticed me to buy them. The book was sent to me by the JASA book editor for review and I went through the whole of it during my flight back from Jeddah. [Disclaimer about potential self-plagiarism: this post or a likely edited version of it will eventually appear in JASA. If not CHANCE, for once.]

The very first sentence (after the preface) quotes my late friend Steve Fienberg, which is definitely starting on the right foot. The exposition of the motivations for writing the book is quite convincing, with more emphasis than usual put on the notion and limitations of modeling. The discourse is overall inspirational and contains many relevant remarks and links that make it worth reading it as a whole. While heavily connected with a few R packages like fitdist, fitistrplus, brms (a  front for Stan), glm, glmer, the book is wisely bypassing the perilous reef of recalling R bases. Similarly for the foundations of probability and statistics. While lacking in formal definitions, in my opinion, it reads well enough to somehow compensate for this very lack. I also appreciate the coherent and throughout continuation of the parallel description of Bayesian and non-Bayesian analyses, an attempt that often too often quickly disappear in other books. (As an aside, note that hardly anyone claims to be a frequentist, except maybe Deborah Mayo.) A new model is almost invariably backed by a new dataset, if a few being somewhat inappropriate as in the mammal sleep patterns of Chapter 5. Or in Fig. 6.1.

Given that the main motivation for the book (when compared with references like BDA) is heavily towards the practical implementation of statistical modelling via R packages, it is inevitable that a large fraction of Statistical Modeling with R is spent on the analysis of R outputs, even though it sometimes feels a wee bit too heavy for yours truly.  The R screen-copies are however produced in moderate quantity and size, even though the variations in typography/fonts (at least on my copy?!) may prove confusing. Obviously the high (explosive?) distinction between regression models may eventually prove challenging for the novice reader. The specific issue of prior input (or “defining priors”) is briefly addressed in a non-chapter (p.323), although mentions are made throughout preceding chapters. I note the nice appearance of hierarchical models and experimental designs towards the end, but would have appreciated some discussions on missing topics such as time series, causality, connections with machine learning, non-parametrics, model misspecification. As an aside, I appreciated being reminded about the apocryphal nature of Ockham’s much cited quote “Pluralitas non est ponenda sine necessitate“.

Typo Jeffries found in Fig. 2.1, along with a rather sketchy representation of the history of both frequentist and Bayesian statistics. And Jon Wakefield’s book (with related purpose of presenting both versions of parametric inference) was mistakenly entered as Wakenfield’s in the bibliography file. Some repetitions occur. I do not like the use of the equivalence symbol ≈ for proportionality. And I found two occurrences of the unavoidable “the the” typo (p.174 and p.422). I also had trouble with some sentences like “long-run, hypothetical distribution of parameter estimates known as the sampling distribution” (p.27), “maximum likelihood estimates [being] sufficient” (p.28), “Jeffreys’ (1939) conjugate priors” [which were introduced by Raiffa and Schlaifer] (p.35), “A posteriori tests in frequentist models” (p.130), “exponential families [having] limited practical implications for non-statisticians” (p.190), “choice of priors being correct” (p.339), or calling MCMC sample terms “estimates” (p.42), and issues with some repetitions, missing indices for acronyms, packages, datasets, but did not bemoan the lack homework sections (beyond suggesting new datasets for analysis).

A problematic MCMC entry is found when calibrating the choice of the Metropolis-Hastings proposal towards avoiding negative values “that will generate an error when calculating the log-likelihood” (p.43) since it suggests proposed values should not exceed the support of the posterior (and indicates a poor coding of the log-likelihood!). I also find the motivation for the full conditional decomposition behind the Gibbs sampler (p.47) unnecessarily confusing. (And automatically having a Metropolis-Hastings step within Gibbs as on Fig. 3.9 brings another magnitude of confusion.) The Bayes factor section is very terse. The derivation of the Kullback-Leibler representation (7.3) as an expected log likelihood ratio seems to be missing a reference measure. Of course, seeing a detailed coverage of DIC (Section 7.4) did not suit me either, even though the issue with mixtures was alluded to (with no detail whatsoever). The Nelder presentation of the generalised linear models felt somewhat antiquated, since the addition of the scale factor a(φ) sounds over-parameterized.

But those are minor quibble in relation to a book that should attract curious minds of various background knowledge and expertise in statistics, as well as work nicely to support an enthusiastic teacher of statistical modelling. I thus recommend this book most enthusiastically.

Measuring abundance [book review]

This 2020 book, Measuring Abundance:  Methods for the Estimation of Population Size and Species Richness was written by Graham Upton, retired professor of applied statistics, for the Data in the Wild series published by Pelagic Publishing, a publishing company based in Exeter.

“Measuring the abundance of individuals and the diversity of species are core components of most ecological research projects and conservation monitoring. This book brings together in one place, for the first time, the methods used to estimate the abundance of individuals in nature.”

Its purpose is to provide a collection of statistical methods for measuring animal abundance or lack thereof. There are four parts: a primer on statistical methods, going no further than maximum likelihood estimation and bootstrap. The term Bayesian only occurs once, in connection with the (a-Bayesian) BIC. (I first spotted a second entry, until I realised this was not a typo and the example truly was about Bawean warty pigs!) The second part is about stationary (or static) individuals, such as trees, and it mostly exposes different recognised ways of sampling, with a focus on minimising the surveyor’s effort. Examples include forestry sampling (with a chainsaw method!) and underwater sampling. There is very little statistics involved in this part apart from the rare appearance of a MLE with an asymptotic confidence interval. There is also very little about misspecified models, except for the occasional warning that the estimates may prove completely wrong. The third part is about mobile individuals, with capture-recapture methods receiving the lion’s share (!). No lion was actually involved in the studies used as examples (but there were grizzly bears from Yellowstone and Banff National Parks). Given the huge variety of capture-recapture models, very little input is found within the book as the practical aspects are delegated to R software like the RMark and mra packages. Very little is written on using covariates or spatial features in such models, mostly dedicated to printed output from R packages with AIC as the sole standard for comparing models. I did not know of distance methods (Chapter 8), which are less invasive counting methods. They however seem to rely on a particular model of missing on individuals as the distance increases. The last section is about estimating the number of species. With again a model assumption that may prove wrong. With the inclusion of diversity measures,

The contents of the book are really down to earth and intended for field data gatherers. For instance, “drive slowly and steadily at 20 mph with headlights and hazard lights on ” (p.91) or “Before starting to record, allow fish time to acclimatize to the presence of divers” (p.91). It is unclear to me how useful the book would prove to be for general statisticians, apart from revealing the huge diversity of methods actually employed in the field. To either build upon these or expose students to their reassessment. More advanced books are McCrea and Morgan (2014), Buckland et al. (2016) and the most recent Seber and Schofield (2019).

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

mathematical theory of Bayesian statistics [book review]

I came by chance (and not by CHANCE) upon this 2018 CRC Press book by Sumio Watanabe and ordered it myself to gather which material it really covered. As the back-cover blurb was not particularly clear and the title sounded quite general. After reading it, I found out that this is a mathematical treatise on some aspects of Bayesian information criteria, in particular on the Widely Applicable Information Criterion (WAIC) that was introduced by the author in 2010. The result is a rather technical and highly focussed book with little motivation or intuition surrounding the mathematical results, which may make the reading arduous for readers. Some background on mathematical statistics and Bayesian inference is clearly preferable and the book cannot be used as a textbook for most audiences, as opposed to eg An Introduction to Bayesian Analysis by J.K. Ghosh et al. or even more to Principles of Uncertainty by J. Kadane. In connection with this remark the exercises found in the book are closer to the delivery of additional material than to textbook-style exercises.

“posterior distributions are often far from any normal distribution, showing that Bayesian estimation gives the more accurate inference than other estimation methods.”

The overall setting is one where both the sampling and the prior distributions are different from respective “true” distributions. Requiring a tool to assess the discrepancy when utilising a specific pair of such distributions. Especially when the posterior distribution cannot be approximated by a Normal distribution. (Lindley’s paradox makes an interesting incognito incursion on p.238.) The WAIC is supported for the determination of the “true” model, in opposition to AIC and DIC, incl. on a mixture example that reminded me of our eight versions of DIC paper. In the “Basic Bayesian Theory” chapter (§3), the “basic theorem of Bayesian statistics” (p.85) states that the various losses related with WAIC can be expressed as second-order Taylor expansions of some cumulant generating functions, with order o(n⁻¹), “even if the posterior distribution cannot be approximated by any normal distribution” (p.87). With the intuition that

“if a log density ratio function has a relatively finite variance then the generalization loss, the cross validation loss, the training loss and WAIC have the same asymptotic behaviors.”

Obviously, these “basic” aspects should come as a surprise to a fair percentage of Bayesians (in the sense of not being particularly basic). Myself included. Chapter 4 exposes why, for regular models, the posterior distribution accumulates in an ε neighbourhood of the optimal parameter at a speed O(n^2/5). With the normalised partition function being of order n^-d/2 in the neighbourhood and exponentially negligible outside. A consequence of this regular asymptotic theory is that all above losses are asymptotically equivalent to the negative log likelihood plus similar order n⁻¹ terms that can be ordered. Chapters 5 and 6 deal with “standard” [the likelihood ratio is a multi-index power of the parameter ω] and general posterior distributions that can be written as mixtures of standard distributions,  with expressions of the above losses in terms of new universal constants. Again, a rather remote concern of mine. The book also includes a chapter (§7) on MCMC, with a rather involved proof that a Metropolis algorithm satisfies detailed balance (p.210). The Gibbs sampling section contains an extensive example on a two-dimensional two-component unit-variance Normal mixture, with an unusual perspective on the posterior, which is considered as “singular” when the true means are close. (Label switching or the absence thereof is not mentioned.) In terms of approximating the normalising constant (or free energy), the only method discussed there is path sampling, with a cryptic remark about harmonic mean estimators (not identified as such). In a final knapsack chapter (§9),  Bayes factors (confusedly denoted as L(x)) are shown to be most powerful tests in a Bayesian sense when comparing hypotheses without prior weights on said hypotheses, while posterior probability ratios are the natural statistics for comparing models with prior weights on said models. (With Lindley’s paradox making another appearance, still incognito!) And a  notion of phase transition for hyperparameters is introduced, with the meaning of a radical change of behaviour at a critical value of said hyperparameter. For instance, for a simple normal- mixture outlier model, the critical value of the Beta hyperparameter is α=2. Which is a wee bit of a surprise when considering Rousseau and Mengersen (2011) since their bound for consistency was α=d/2.

In conclusion, this is quite an original perspective on Bayesian models, covering the somewhat unusual (and potentially controversial) issue of misspecified priors and centered on the use of information criteria. I find the book could have benefited from further editing as I noticed many typos and somewhat unusual sentences (at least unusual to me).

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

Lindley’s paradox as a loss of resolution

“The principle of indifference states that in the absence of prior information, all mutually exclusive models should be assigned equal prior probability.”

Colin LaMont and Paul Wiggins arxived a paper on Lindley’s paradox a few days ago. The above quote is the (standard) argument for picking (½,½) partition between the two hypotheses, which I object to if only because it does not stand for multiple embedded models. The main point in the paper is to argue about the loss of resolution induced by averaging against the prior, as illustrated by the picture above for the N(0,1) versus N(μ,1) toy problem. What they call resolution is the lowest possible mean estimate for which the null is rejected by the Bayes factor (assuming a rejection for Bayes factors larger than 1). While the detail is missing, I presume the different curves on the lower panel correspond to different choices of L when using U(-L,L) priors on μ… The “Bayesian rejoinder” to the Lindley-Bartlett paradox (p.4) is in tune with my interpretation, namely that as the prior mass under the alternative gets more and more spread out, there is less and less prior support for reasonable values of the parameter, hence a growing tendency to accept the null. This is an illustration of the long-lasting impact of the prior on the posterior probability of the model, because the data cannot impact the tails very much.

“If the true prior is known, Bayesian inference using the true prior is optimal.”

This sentence and the arguments following is meaningless in my opinion as knowing the “true” prior makes the Bayesian debate superfluous. If there was a unique, Nature provided, known prior π, it would loose its original meaning to become part of the (frequentist) model. The argument is actually mostly used in negative, namely that since it is not know we should not follow a Bayesian approach: this is, e.g., the main criticism in Inferential Models. But there is no such thing as a “true” prior! (Or a “true’ model, all things considered!) In the current paper, this pseudo-natural approach to priors is utilised to justify a return to the pseudo-Bayes factors of the 1990’s, when one part of the data is used to stabilise and proper-ise the (improper) prior, and a second part to run the test per se. This includes an interesting insight on the limiting cases of partitioning corresponding to AIC and BIC, respectively, that I had not seen before. With the surprising conclusion that “AIC is the derivative of BIC”!

model selection and multiple testing

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.