Large-scale Inference by Brad Efron is the first IMS Monograph in this new series, coordinated by David Cox and published by Cambridge University Press. Since I read this book immediately after Cox’ and Donnelly’s Principles of Applied Statistics, I was thinking of drawing a parallel between the two books. However, while none of them can be classified as textbooks [even though Efron's has exercises], they differ very much in their intended audience and their purpose. As I wrote in the review of Principles of Applied Statistics, the book has an encompassing scope with the goal of covering all the methodological steps required by a statistical study. In Large-scale Inference, Efron focus on empirical Bayes methodology for large-scale inference, by which he mostly means multiple testing (rather than, say, data mining). As a result, the book is centred on mathematical statistics and is more technical. (Which does not mean it less of an exciting read!) The book was recently reviewed by Jordi Prats for Significance. Akin to the previous reviewer, and unsurprisingly, I found the book nicely written, with a wealth of R (colour!) graphs (the R programs and dataset are available on Brad Efron’s home page).
“I have perhaps abused the “mono” in monograph by featuring methods from my own work of the past decade.” (p.xi)
Sadly, I cannot remember if I read my first Efron’s paper via his 1977 introduction to the Stein phenomenon with Carl Morris in Pour la Science (the French translation of Scientific American) or through his 1983 Pour la Science paper with Persi Diaconis on computer intensive methods. (I would bet on the later though.) In any case, I certainly read a lot of the Efron’s papers on the Stein phenomenon during my thesis and it was thus with great pleasure that I saw he introduced empirical Bayes notions through the Stein phenomenon (Chapter 1). It actually took me a while but I eventually (by page 90) realised that empirical Bayes was a proper subtitle to Large-Scale Inference in that the large samples were giving some weight to the validation of empirical Bayes analyses. In the sense of reducing the importance of a genuine Bayesian modelling (even though I do not see why this genuine Bayesian modelling could not be implemented in the cases covered in the book).
“Large N isn’t infinity and empirical Bayes isn’t Bayes.” (p.90)
The core of Large-scale Inference is multiple testing and the empirical Bayes justification/construction of Fdr’s (false discovery rates). Efron wrote more than a dozen papers on this topic, covered in the book and building on the groundbreaking and highly cited Series B 1995 paper by Benjamini and Hochberg. (In retrospect, it should have been a Read Paper and so was made a “retrospective read paper” by the Research Section of the RSS.) Frd are essentially posterior probabilities and therefore open to empirical Bayes approximations when priors are not selected. Before reaching the concept of Fdr’s in Chapter 4, Efron goes over earlier procedures for removing multiple testing biases. As shown by a section title (“Is FDR Control “Hypothesis Testing”?”, p.58), one major point in the book is that an Fdr is more of an estimation procedure than a significance-testing object. (This is not a surprise from a Bayesian perspective since the posterior probability is an estimate as well.)
“Scientific applications of single-test theory most often suppose, or hope for rejection of the null hypothesis (…) Large-scale studies are usually carried out with the expectation that most of the N cases will accept the null hypothesis.” (p.89)
On the innovations proposed by Efron and described in Large-scale Inference, I particularly enjoyed the notions of local Fdrs in Chapter 5 (essentially pluggin posterior probabilities that a given observation stems from the null component of the mixture) and of the (Bayesian) improvement brought by empirical null estimation in Chapter 6 (“not something one estimates in classical hypothesis testing”, p.97) and the explanation for the inaccuracy of the bootstrap (which “stems from a simpler cause”, p.139), but found less crystal-clear the empirical evaluation of the accuracy of Fdr estimates (Chapter 7, ‘independence is only a dream”, p.113), maybe in relation with my early career inability to explain Morris’s (1983) correction for empirical Bayes confidence intervals (pp. 12-13). I also discovered the notion of enrichment in Chapter 9, with permutation tests resembling some low-key bootstrap, and multiclass models in Chapter 10, which appear as if they could benefit from a hierarchical Bayes perspective. The last chapter happily concludes with one of my preferred stories, namely the missing species problem (on which I hope to work this very Spring).
Several interesting questions were raised during my seminar talk at Oxford. First, David Cox suggested I looked at the collection of the two p-values in the Poisson and geometric cases to check whether or not they could point out at a disagreement. (I am however unsure at how the p-values should be computed in this case, maybe as a likelihood-ratio test…) Chris Holmes asked about what happened to the ABC Bayes factor when both models were wrong. I had not thought of this earlier and will look into it: my first impression is that there is no reason for the same model to be chosen. It depends on the relative tail behaviour of the distribution of the summary statistics under both models… Stephen Lauritzen mentioned prior to the seminar a highly relevant book by a Copenhagen mathematician, whose definition of conditional densities was perfectly suited for constructing a convergence proof for ABC (to be incorporated in my Roma slides, if feasible). During the talk, he also pointed out at other (counter)examples of models where sufficient statistics remained sufficient across models: e.g., contingency tables with pairwise interactions. Arnaud Doucet got back to the Potts model (counter)example to stress that we needed perfect sampling to make it work and that our MCMC alternative could be adding another level of approximation to the process, which is quite right!
