Philosophies, Puzzles, and Paradoxes: A Statistician's Search for the TruthYudiPawitan and YoungjoLeeChapman & Hall/CRC, 2024, xiv + 351 pages, £18.39/$23.96 paperback, £104/$136 hardback, £17.24/$22.46 eBook ISBN: 9781032377391 paperback; 978103

Readership: Statistics graduate and advanced undergraduate students, practitioners, philosophers of science, and general readers intrigued by puzzles and paradoxes. A comment in the preface gives a fair indication of the character of this book: Our book is an eclectic, some might even say idiosyncratic, selection of topics that reflect our interests. Perhaps, inevitably, this review will have some of that same character in the points chosen to highlight. The book is divided into four parts. Part I (Chapters 1–4) treats ‘How Do We Know Whether What We Believe Is True’; Part II (5–9) treats ‘Probability and Inverse-Probability Inference’. Inverse probability is what is now called Bayesian inference; Part III (10–13) treats ‘Likelihood and Likelihood-Based Inference’; and Part IV (14–19) treats ‘Puzzles and Paradoxes’ where, in a number of instances, the authors take up paradoxes that have been discussed earlier in the text again. A list that appears prior to the introduction names 30 paradoxes or paradox variations. Chapter 1 begins with a quotation from Edmond Burke (1729–1797): Whoever undertakes to set himself up as a judge of Truth and Knowledge is shipwrecked by the laughter of the gods. Rationalism (as espoused by Descartes, Leibnitz, Kant, and Hegel) is contrasted with empiricism (Aristotle, Locke and Hume), and the logical positivism of the Vienna circle. The distinctions are, the authors acknowledge, in large part a matter of emphasis. Chapter 2 moves, with the same attention to the historical background, to contrast deduction with induction, pointing out their complementary roles in advancing scientific knowledge. Chapter 3 points out the limits of synthetic a priori knowledge, drawing attention to Gödel's work that showed that in any axiomatic system that is rich enough to cover basic arithmetic, there will be propositions that can neither be proved nor disproved. Chapter 4's account of how ‘real’ scientific processes work, with a constant interplay between thinking and observation, focuses mainly on the importance, at least in some contexts, of paradigm shifts. Continental drift, proposed by Wegener in 1912, was not widely accepted until incontrovertible evidence for plate tectonics emerged in the 1960s. Several other examples are given. Later chapters do not, as one might have expected, build in any substantive way on this introductory and selective account of ‘real’ scientific processes. Part II (Chapters 5–9) moves on to issues of probability and probabilistic inference. The seventeenth century saw the beginnings of the modern theory of probability, focusing on the analysis of games of chance. With Kolmogorov in 1933, the mathematical theory of probability was placed on a sound axiomatic basic. The authors argue that for serious students of statistics, the philosophy is as important as the mathematics of probability. They note that there is not just one philosophy, but competing philosophies, with the main division is between the frequentist and Bayesian schools. Chapter 6 follows Gillies (2012) in describing a finer division into six different schools. Chapter 7 has a detailed and technical discussion of Savage's (1972) axioms, seen as a strong influence on modern Bayesian approaches, which are then treated in Chapters 8 and 9. The authors discuss, primarily, ‘non-subjective non-informative priors’, with particular attention to the Jeffreys' prior, noting its difficulties for extension to the multi-parameter case. The best one can hope for, the authors say, ‘seems to be that the prior should not be dominant relative to the data in determining the posterior’. On page 89 the authors say Most philosophers appear to remain unconvinced about whether we can have an objective probability for single events. Then on page 195, we find the stronger statement: It is a longstanding question in the history of probability whether there is an objective probability for single events. … The logical difficulties, mostly associated with the reference class, are reviewed on page 87. Because of these difficulties, frequentists simply reject the notions of such a probability. There are frequent probability statements in the text that relate to single events, some related to contexts where there is a physical chance mechanism and some where the long-term probability is given by a hypothetical statistical model. If the issue is one of single as opposed to more than one event, why should the addition of one or more events overcome the reference class issue? The strict frequentist objection is surely to the use of subjective prior probabilities that are unsupported by empirical evidence. As pointed out in Wit et al. (2012), the choice of model is a reference problem that applies to all flavours of inference—frequentist, Bayesian, and the claimed ‘non-subjective’ inference that the authors present as underpinning ideas of ‘epistemic confidence’. Paradoxes, discussed at various places in Parts I to III as well as having Part IV to themselves, do appear to be the primary focus of the book, with the other content providing a context for their discussion. For most of these, there is a Wikipedia article that can often helpfully supplement the account given in this book. As one example, consider the ‘exchange paradox’, discussed on pages 154 and 283, with several variants discussed on other pages. For this, an unknown θ dollars is placed in one envelope and 2θ dollars in another, and it is assumed that for all practical purposes, all values of θ are equally likely. The following is adapted from the authors' account: You pick one envelope at random, and open it to reveal y = $100. You are then asked whether you would like to exchange it for the other envelope. You reason that the other envelope will either contain $50 or $200, with equal probability. If you exchange the envelope, the expected value of your winning is $(50 + 100)/2 = $125, a gain of $25. In fact, there will be such a ‘gain’, of (y/2 + 2y)/2 − y = y/4, irrespective of the amount y in the envelope—you do not have to open the envelope to know this. This makes no sense. The same argument would imply that one can swap again and again, and gain the relevant y/4 each time! The argument is made that the outcomes $50 and $200 have equal likelihood, not equal probability. Taking x to be the smaller of the two amounts, probabilities for y that do make sense are for y = x and y = 2x, with equal probability. There will either be a gain of x by swapping, or a loss of x, so that the change averages out to (x - x) = 0. Several different types of paradoxes of human choice are discussed— choices that, from the perspective of the Expected Utility Theory of classical economics, are irrational. These feature in Kahneman (2011), who with Tversky developed Prospect Theory as a model for actual human behaviour—work that was cited in Kahneman's 2002 award of a Nobel prize in economics. Under the heading of ‘Fallacious Fallacies’, the authors treat ‘fallacies’ that may in reality reflect different uses of the word likely or probable, on the one hand to mean ‘better identifies’ and on the other in a strict probability sense. The authors note (page 167) that ‘significance testing and the P-value are perhaps the most dominant in scientific research’, later commenting (page 169) that ‘there are dissenting views about its utility’. They go on to state (page 205), in the context of a discussion of ‘Epistemic Confidence and Objective Bayesianism’: The confidence distribution for normal means is the same as the Bayesian posterior under a uniform prior. This is then the basis for the claim that epistemic confidence ‘can be considered a non-Bayesian alternative to the Bayesian posterior’. As an unbounded uniform prior rarely makes sense, it would seem that this places strong limits on the use of epistemic confidence in inference. Even for the smallest sample sizes, the data overwhelms this choice of prior. p $$ p $$ -value fallacies are not highlighted. Specifically, the 0.05 probability applies when 0.05 is specified in advance as a threshold, not when p = 0.05 $$ p=0.05 $$ . More importantly, the reference class to which p ≤ 0.05 $$ p\le 0.05 $$ applies is not in a significance test context that for the alternative against the null, and does not imply a 19:1 odds ratio in favour of the alternative. An argument can be made that p = 0.05 $$ p=0.05 $$ translates into a posterior odds of no more than 2.5 R $$ 2.5R $$ , where R $$ R $$ is the prior odds ratio, but all depends on the unknown probability of a false positive under the alternative. See Benjamin & Berger (2019) for commentary. Nor is there any discussion of the way that the issues and ideas that are discussed feed through into scientific research and publication processes. One result of the widespread use of a p ≤ 0.05 $$ p\le 0.05 $$ criterion in deciding what gets published is to inflate effect sizes in what is published. Reproducibility studies, perhaps better called replication studies, have been important windows into the issues that arise in many areas of laboratory studies, and have led to demands for changes in publication practices. For the 97 studies reported in Open Science Collaboration (2015), the average amount by which the Cohen's d effect size in the original study exceeded the effect size in the replication was 0.46, with a standard error equal to 0.066. Comparable differences have been reported in replication studies in, among other areas, pre-clinical cancer research and laboratory economics. Replication studies shed important light on the scientific processes that lie behind what gets reported, and raise questions that should not be ignored. This is a book for those who want to explore beyond such more systematic accounts of the philosophy of probability as Gillies (2012), and especially for those who want to probe often subtle paradoxes that can help develop statistical intuition. As Kahneman (2011) has commented, humans are poor intuitive statisticians. In that respect, a useful supplement would be R or other such code for simulations that illustrate the paradoxes.