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.]

ABC model choice by random forests [guest post]

[Dennis Prangle sent me his comments on our ABC model choice by random forests paper. Here they are! And I appreciate very much contributors commenting on my paper or others, so please feel free to join.]

This paper proposes a new approach to likelihood-free model choice based on random forest classifiers. These are fit to simulated model/data pairs and then run on the observed data to produce a predicted model. A novel “posterior predictive error rate” is proposed to quantify the degree of uncertainty placed on this prediction. Another interesting use of this is to tune the threshold of the standard ABC rejection approach, which is outperformed by random forests.

The paper has lots of thought-provoking new ideas and was an enjoyable read, as well as giving me the encouragement I needed to read another chapter of the indispensable Elements of Statistical Learning However I’m not fully convinced by the approach yet for a few reasons which are below along with other comments.

Alternative schemes

The paper shows that random forests outperform rejection based ABC. I’d like to see a comparison to more efficient ABC model choice algorithms such as that of Toni et al 2009. Also I’d like to see if the output of random forests could be used as summary statistics within ABC rather than as a separate inference method.

Posterior predictive error rate (PPER)

This is proposed to quantify the performance of a classifier given a particular data set. The PPER is the proportion of times the classifier’s most favoured model is incorrect for simulated model/data pairs drawn from an approximation to the posterior predictive. The approximation is produced by a standard ABC analysis.

Misclassification could be due to (a) a poor classifier or (b) uninformative data, so the PPER aggregrates these two sources of uncertainty. I think it is still very desirable to have an estimate of the uncertainty due to (b) only i.e. a posterior weight estimate. However the PPER is useful. Firstly end users may sometimes only care about the aggregated uncertainty. Secondly relative PPER values for a fixed dataset are a useful measure of uncertainty due to (a), for example in tuning the ABC threshold. Finally, one drawback of the PPER is the dependence on an ABC estimate of the posterior: how robust are the results to the details of how this is obtained?

Classification

This paper illustrates an important link between ABC and machine learning classification methods: model choice can be viewed as a classification problem. There are some other links: some classifiers make good model choice summary statistics (Prangle et al 2014) or good estimates of ABC-MCMC acceptance ratios for parameter inference problems (Pham et al 2014). So the good performance random forests makes them seem a generally useful tool for ABC (indeed they are used in the Pham et al al paper).

Advances in scalable Bayesian computation [day #2]

And here is the second day of our workshop Advances in Scalable Bayesian Computation gone! This time, it sounded like the “main” theme was about brains… In fact, Simon Barthelmé‘s research originated from neurosciences, while Dawn Woodard dissected a brain (via MRI) during her talk! (Note that the BIRS website currently posts Simon’s video as being Dan Simpson’s talk, the late change in schedule being due to Dan most unfortunately losing his passport during a plane transfer and most unfortunately being prevented from attending…) I found Simon’s talk quite inspiring, with this Tibshirani et al.’s trick of using logistic regression to estimate densities as a classification problem central to the method and suggesting a completely different vista for handling normalising constants… Then Raazesh Sainudiin gave a detailed explanation and validation of his approach to density estimation by multidimensional pavings/histograms, with a tree representation allowing for fast merging of different estimators. Raaz had given a preliminary version of the talk at CREST last Fall, which helped with focussing on the statistical aspects of the method. Chris Strickland then exposed an image analysis of flooded Northern Queensland landscapes, using a spatio-temporal model with changepoints and about 18,000 parameters. still managing to get an efficiency of O(np) thanks to two tricks. Then it was time for the group photograph outside in a balmy -18⁰ and an open research time that was quite profitable.

In the afternoon sessions, Paul Fearnhead presented an auxiliary variable approach to particle Gibbs, which again opened new possibilities for handling state-space models, but also reminding me of Xiao-Li Meng’s reparameterisation devices. And making me wonder (out loud) whether or not the SMC algorithm was that essential in a static setting, since the sequence could be explored in any possible order for a fixed time horizon. Then Emily Fox gave a 2-for-1 talk, mostly focussing on the first talk, where she introduced a new technique for approximating the gradient in Hamiltonian (or Hockey!) Monte Carlo, using second order Langevin. She did not have much time for the second talk, which intersected with the one she gave at BNP’ski in Chamonix, but focussed on a notion of sandwiched slice sampling where the target density only needs bounds that can get improved if needed. A cool trick! And the talks ended with Dawn Woodard‘s analysis of time varying 3-D brain images towards lesion detection, through an efficient estimation of a spatial mixture of normals.