Gaussian process (GP) regression is computationally expensive in spatial applications involving massive data. Various methods address this limitation, including a small number of Bayesian methods based on distributed computations (or the divide-and-conquer strategy). Focusing on the latter literature, we achieve three main goals. First, we develop an extensible Bayesian framework for distributed spatial GP regression that embeds many popular methods. The proposed framework has three steps that partition the entire data into many subsets, apply a readily available Bayesian spatial process model in parallel on all the subsets, and combine the posterior distributions estimated on all the subsets into a pseudo posterior distribution that conditions on the entire data. The combined pseudo posterior distribution replaces the full data posterior distribution in prediction and inference problems. Demonstrating our framework’s generality, we extend posterior computations for (nondistributed) spatial process models with a stationary full-rank and a nonstationary low-rank GP priors to the distributed setting. Second, we contrast the empirical performance of popular distributed approaches with some widely-used, nondistributed alternatives and highlight their relative advantages and shortcomings. Third, we provide theoretical support for our numerical observations and show that the Bayes ${\mathit{L}}_2$-risks of the combined posterior distributions obtained from a subclass of the divide-and-conquer methods achieves the near-optimal convergence rate in estimating the true spatial surface with various types of covariance functions. Additionally, we provide upper bounds on the number of subsets to achieve these near-optimal rates.

The development of John Aitchison’s approach to compositional data analysis is followed since his paper read to the Royal Statistical Society in 1982. Aitchison’s logratio approach, which was proposed to solve the problematic aspects of working with data with a fixed-sum constraint, is summarized and reappraised. It is maintained that the properties on which this approach was originally built, the main one being subcompositional coherence, are not required to be satisfied exactly—quasi-coherence is sufficient, that is near enough to being coherent for all practical purposes. This opens up the field to using simpler data transformations, such as power transformations, that permit zero values in the data. The additional property of exact isometry, which was subsequently introduced and not in Aitchison’s original conception, imposed the use of isometric logratio transformations, but these are complicated and problematic to interpret, involving ratios of geometric means. If this property is regarded as important in certain analytical contexts, for example, unsupervised learning, it can be relaxed by showing that regular pairwise logratios, as well as the alternative quasi-coherent transformations, can also be quasi-isometric, meaning they are close enough to exact isometry for all practical purposes. It is concluded that the isometric and related logratio transformations such as pivot logratios are not a prerequisite for good practice, although many authors insist on their obligatory use. This conclusion is fully supported here by case studies in geochemistry and in genomics, where the good performance is demonstrated of pairwise logratios, as originally proposed by Aitchison, or Box–Cox power transforms of the original compositions where no zero replacements are necessary.

This paper takes the reader on a journey through the history of Bayesian computation, from the 18th century to the present day. Beginning with the one-dimensional integral first confronted by Bayes in 1763, we highlight the key contributions of: Laplace, Metropolis (and, importantly, his coauthors), Hammersley and Handscomb, and Hastings, all of which set the foundations for the computational revolution in the late 20th century—led, primarily, by Markov chain Monte Carlo (MCMC) algorithms. A very short outline of 21st century computational methods—including pseudo-marginal MCMC, Hamiltonian Monte Carlo, sequential Monte Carlo and the various “approximate” methods—completes the paper.

Classical Randomized Controlled Trials (RCTs), or A/B tests, are designed to draw causal inferences about a population of units, for example, individuals, plots of land or visits to a website. A key assumption underlying a standard RCT is the absence of interactions between units, or the stable unit treatment value assumption (Ann. Statist. 6 (1978) 34–58). Modern experimentation, however, is often conducted in settings characterized by complex interactions between units. Such interactions can invalidate the standard estimators and make classical experimental designs ineffective. Although the presence of interference forces us to make untestable assumptions on the nature of the interactions even under randomization, sophisticated experimental designs can ameliorate the dependence on such assumptions. In this manuscript, we review the recent and rapidly growing literature on novel experimental designs for these settings. One key feature common to many of these designs is the presence of multiple layers of randomization within the same experiment. We discuss a novel experimental design, called Multiple Randomization Designs or MRDs, that provides a general framework for such experiments. Through these complex designs, we can study questions about causal effects in the presence of interference that cannot be answered by classical RCTs.

In an observational study of the effects caused by a treatment, biases from unmeasured covariates remain a concern even after successful adjustments for measured covariates. This concern is partly addressed by demonstrating that the qualitative conclusions of the primary analysis would not be altered by small or moderate biases—that these conclusions are insensitive to small or moderate bias. Additionally, the concern is partly addressed by collecting additional information, such as outcomes known to be unaffected by the treatment, and using this information as a test of various biases. Is there a gap between these two activities? Perhaps the study is insensitive to small biases, and we can detect large biases, but the study is sensitive to moderate biases that cannot be detected—that is an informal description of a gap. The concept of “no gap” is defined formally in Definition 3.1, and the probability of “no gap” is determined under various sampling situations. When there is no gap, ask: Are causal conclusions measurably strengthened? If so, by how much? The answer depends upon the covering design sensitivity, $\stackrel{⌢}{\mathrm{\Gamma }}$, defined to be the smallest bias that can explain both the ostensible effect of the treatment on the primary outcome and the evidence of bias provided by the unaffected outcome. The covering design sensitivity is calculated in various contexts. A small observational study of the effects of light alcohol consumption on HDL cholesterol is used to illustrate ideas and methods.

Given a sequence $\mathit{X}=({\mathit{X}}_1,{\mathit{X}}_2,\dots )$ of random observations, a Bayesian forecaster aims to predict ${\mathit{X}}_{\mathit{n}+1}$ based on $({\mathit{X}}_1,\dots ,{\mathit{X}}_{\mathit{n}})$ for each $\mathit{n}\ge 0$. To this end, in principle, she only needs to select a collection $\mathit{\sigma }=({\mathit{\sigma }}_0,{\mathit{\sigma }}_1,\dots )$, called “strategy” in what follows, where ${\mathit{\sigma }}_0(·)=\mathit{P}({\mathit{X}}_1\in ·)$ is the marginal distribution of ${\mathit{X}}_1$ and ${\mathit{\sigma }}_{\mathit{n}}(·)=\mathit{P}({\mathit{X}}_{\mathit{n}+1}\in ·|{\mathit{X}}_1,\dots ,{\mathit{X}}_{\mathit{n}})$ the nth predictive distribution. Because of the Ionescu–Tulcea theorem, σ can be assigned directly, without passing through the usual prior/posterior scheme. One main advantage is that no prior probability is to be selected. In a nutshell, this is the predictive approach to Bayesian learning. A concise review of the latter is provided in this paper. We try to put such an approach in the right framework, to make clear a few misunderstandings, and to provide a unifying view. Some recent results are discussed as well. In addition, some new strategies are introduced and the corresponding distribution of the data sequence X is determined. The strategies concern generalized Pólya urns, random change points, covariates and stationary sequences.

The 21st century has seen an enormous growth in the development and use of approximate Bayesian methods. Such methods produce computational solutions to certain “intractable” statistical problems that challenge exact methods like Markov chain Monte Carlo: for instance, models with unavailable likelihoods, high-dimensional models and models featuring large data sets. These approximate methods are the subject of this review. The aim is to help new researchers in particular—and more generally those interested in adopting a Bayesian approach to empirical work—distinguish between different approximate techniques, understand the sense in which they are approximate, appreciate when and why particular methods are useful and see the ways in which they can can be combined.

There has been an intense recent activity in embedding of very high-dimensional and nonlinear data structures, much of it in the data science and machine learning literature. We survey this activity in four parts. In the first part, we cover nonlinear methods such as principal curves, multidimensional scaling, local linear methods, ISOMAP, graph-based methods and diffusion mapping, kernel based methods and random projections. The second part is concerned with topological embedding methods, in particular mapping topological properties into persistence diagrams and the Mapper algorithm. Another type of data sets with a tremendous growth is very high-dimensional network data. The task considered in part three is how to embed such data in a vector space of moderate dimension to make the data amenable to traditional techniques such as cluster and classification techniques. Arguably, this is the part where the contrast between algorithmic machine learning methods and statistical modeling, represented by the so-called stochastic block model, is at its greatest. In the paper, we discuss the pros and cons for the two approaches. The final part of the survey deals with embedding in ${\mathbb{R}}^2$, that is, visualization. Three methods are presented: t-SNE, UMAP and LargeVis based on methods in parts one, two and three, respectively. The methods are illustrated and compared on two simulated data sets; one consisting of a triplet of noisy Ranunculoid curves, and one consisting of networks of increasing complexity generated with stochastic block models and with two types of nodes.

Mary E. Thompson (née Beattie) was born September 9, 1944, in Winnipeg, Manitoba, Canada. She obtained a B.Sc. in Mathematics from the University of Toronto in 1965, and earned M.Sc. (1966) and Ph.D. (1969) degrees in Mathematics from the University of Illinois at Urbana-Champaign. She joined the Department of Statistics at the University of Waterloo as a Lecturer in 1969 and became an Assistant Professor in 1971. In 2004, she was awarded the honour of University Professor and in 2011 became Distinguished Professor Emerita at the University of Waterloo. She has served in many leadership roles including Chair of the Department of Statistics and Actuarial Science, Acting Dean of the Faculty of Mathematics, President of the Statistical Society of Canada (SSC) and Chair of the COPSS Presidents’ Award Committee. She chaired the Development Committee for the Canadian Statistical Sciences Institute (CANSSI) and was its founding Scientific Director.

Thompson has received numerous honours and awards including the SSC’s Gold Medal, the Elizabeth L. Scott Award, the Waksberg Award of Survey Methodology and the Governor General’s Innovation Award. She is an elected member of the International Statistical Institute, an Honorary Member of the SSC and is a Fellow of the American Statistical Association, the Institute of Mathematical Statistics, the Royal Society of Canada and the Fields Institute.

Thompson has made fundamental contributions to several areas in statistics including sampling theory and the analysis of surveys. She is the author of two books in these areas: Theory of Sample Surveys (1997) and Sampling Theory and Practices (2020 with C. Wu). She has also made key contributions in estimation theory and stochastic processes. As the author of over 150 published, refereed papers, Thompson has influenced the theory and practice of statistics.

The following conversation took place virtually in September 2022 with interviewer Rhonda J. Rosychuk of the University of Alberta.

Stephen M. Stigler received his Ph.D. in Statistics from the University of California, Berkeley, with a dissertation on the asymptotic distribution of linear functions of order statistics. Starting in 1967, he taught at the University of Wisconsin, Madison, then in 1979 moved to the University of Chicago where he taught from 1979 to 2021. Stigler has worked on a variety of topics in mathematical statistics, ranging from asymptotic theory to the theory of experimental design, and on applications of statistics including in anthropology, forensic science, paleontology, psychology, information transfer and sports. In recent years, he has concentrated on the history of statistics, with inquiries ranging from the development of statistical methods in astronomy and geodesy and their spread to biological and social sciences, to lotteries, to the modern development of statistical theory. He has published four books, The History of Statistics (1986), Statistics on the Table (1999), The Seven Pillars of Statistical Wisdom (2016) and Casanova’s Lottery (2022). A recent research focus has been upon the way the work of Francis Galton on the statistics of inheritance led to the creation of modern multivariate analysis and made a true Bayesian inference possible, and on how R. A. Fisher’s transformation of Karl Pearson’s path breaking research led to a modern period of statistical enlightenment.

Stigler is an elected member of the American Academy of Arts and Sciences and of the American Philosophical Society; he has served as President of the Institute of Mathematical Statistics and of the International Statistical Institute. In 2005, he received the Humboldt Foundation Research Award; in 2010, he was elected Membre Associé of the Académie royale de Belgique, Classe des Sciences. Stigler served as Theory and Methods Editor for the Journal of the American Statistical Association 1979–1982. He was a Guggenheim Fellow in 1977, and received awards for undergraduate teaching at the University of Wisconsin (1971) and University of Chicago (1998).

This interview with Stigler was conducted remotely in July 2021.

Though the notion of exchangeability has been discussed in the causal inference literature under various guises, it has rarely taken its original meaning as a symmetry property of probability distributions. As this property is a standard component of Bayesian inference, we argue that in Bayesian causal inference it is natural to link the causal model, including the notion of confounding and definition of causal contrasts of interest, to the concept of exchangeability. Here, we propose a probabilistic between-group exchangeability property as an identifying condition for causal effects, relate it to alternative conditions for unconfounded inferences (commonly stated using potential outcomes) and define causal contrasts in the presence of exchangeability in terms of posterior predictive expectations for further exchangeable units. While our main focus is on a point treatment setting, we also investigate how this reasoning carries over to longitudinal settings.

I. J. (“Jack”) Good was a leading Bayesian statistician for more than half a century after World War II, playing an important role in the post-war Bayesian revival. But his graduate training had been in pure mathematics rather than statistics (one of his doctoral advisors at Cambridge had been the famous G. H. Hardy). What was responsible for this metamorphosis from pure mathematician to applied and theoretical statistician? As Good himself only revealed in 1976, during the war he had initially served as an assistant to Alan Turing at Bletchley Park, working on the cryptanalysis of the German Naval Enigma, and it was from Turing that he acquired his life-long Bayesian philosophy. Declassified and other documents now permit us to understand in some detail how this came about, and indeed how many of the ideas Good explored and papers he wrote in the initial decades after the war, in fact, gave in sanitized form, results that had their origins in his wartime work. Drawing on these sources, this paper discusses the daily and very real use of Bayesian methods Turing and Good employed, and how this was gradually revealed by Good over the course of his life (including his return to classified work in the 1950s).

In cancer research, clustering techniques are widely used for exploratory analyses, playing a critical role in the identification of novel cancer subtypes and patient management. As data collected by multiple research groups grows, it is increasingly feasible to investigate the replicability of clustering procedures, that is, their ability to consistently recover biologically meaningful clusters across several data sets. In this paper, we review methods for replicability of clustering analyses, and discuss a novel framework for evaluating cross-study clustering replicability, useful when two or more studies are available. Our approach can be applied to any clustering algorithm and can employ different measures of similarity between partitions to quantify replicability, globally (i.e., for the whole sample) as well as locally (i.e., for individual clusters). Using experiments on synthetic and real gene expression data, we illustrate the usefulness of our procedure to evaluate if the same clusters are identified consistently across a collection of data sets.

Using the concept of principal stratification from the causal inference literature, we introduce a new notion of fairness, called principal fairness, for human and algorithmic decision-making. Principal fairness states that one should not discriminate among individuals who would be similarly affected by the decision. Unlike the existing statistical definitions of fairness, principal fairness explicitly accounts for the fact that individuals can be impacted by the decision. This causal fairness formulation also enables online or post-hoc fairness evaluation and policy learning. We also explain how principal fairness relates to the existing causality-based fairness criteria. In contrast to the counterfactual fairness criteria, for example, principal fairness considers the effects of decision in question rather than those of protected attributes of interest. Finally, we discuss how to conduct empirical evaluation and policy learning under the proposed principal fairness criterion.

Gábor J. Székely was born in Budapest, Hungary on February 4, 1947. He graduated from Eötvös Loránd University (ELTE) with an M.S. degree in 1970, and a Ph.D. degree in 1971. He received his Candidate Degree from the Hungarian Academy of Sciences in 1976, and the Doctor of Science Degree (D. Sc.) from the Hungarian Academy of Sciences in 1986. Székely joined the Department of Probability Theory of ELTE in 1970. In 1989, he became the founding chair of the Department of Stochastics of the Budapest Institute of Technology (Technical University of Budapest). In 1995, he moved to the United States as a tenured full professor at Bowling Green State University (BGSU) in Bowling Green, Ohio. Before that, in 1990–1991, he was the first Lukacs Distinguished Professor at BGSU. Székely had several visiting positions, for example, at the University of Amsterdam in 1977 and at Yale University in 1989. Between 2006 and 2022, he served as a Program Director in the Statistics Program of the Division of Mathematical Sciences at the U.S. National Science Foundation. Székely has about 250 publications, including 6 books in several languages. In 1988, he received the Rollo Davidson Prize from Cambridge University, jointly with Imre Z. Ruzsa for their work on algebraic probability theory. In 2010, Székely became an Elected Fellow of the Institute of Mathematical Statistics mostly for his works dealing with physics concepts in statistics like energy statistics and distance correlation. He had the fortune to know and work with world-class mathematicians and statisticians like (in chronological order of their first meetings): P. Erdős, A. Rényi, Y. Linnik, B. de Finetti, A. N. Kolmogorov, H. Robbins, G. Pólya, L. Shepp, G. Wahba, C. R. Rao, B. Efron, P. Bickel and E. Seneta.

We discuss systematically two versions of confidence regions: those based on p-values and those based on e-values, a recent alternative to p-values. Both versions can be applied to multiple hypothesis testing, and in this paper we are interested in procedures that control the number of false discoveries under arbitrary dependence between the base p- or e-values. We introduce a procedure that is based on e-values and show that it is efficient both computationally and statistically using simulated and real-world data sets. Comparison with the corresponding standard procedure based on p-values is not straightforward, but there are indications that the new one performs significantly better in some situations.

Data augmentation improves the convergence of iterative algorithms, such as the EM algorithm and Gibbs sampler by introducing carefully designed latent variables. In this article, we first propose a data augmentation scheme for the first-order autoregression plus noise model, where optimal values of working parameters introduced for recentering and rescaling of the latent states, can be derived analytically by minimizing the fraction of missing information in the EM algorithm. The proposed data augmentation scheme is then utilized to design efficient Markov chain Monte Carlo (MCMC) algorithms for Bayesian inference of some non-Gaussian and nonlinear state space models, via a mixture of normals approximation coupled with a block-specific reparametrization strategy. Applications on simulated and benchmark real data sets indicate that the proposed MCMC sampler can yield improvements in simulation efficiency compared with centering, noncentering and even the ancillarity-sufficiency interweaving strategy.