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.

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.

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.

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.

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.

Statistical modeling can involve a tension between assumptions and statistical identification. The law of the observable data may not uniquely determine the value of a target parameter without invoking a key assumption, and, while plausible, this assumption may not be obviously true in the scientific context at hand. Moreover, there are many instances of key assumptions which are untestable, hence we cannot rely on the data to resolve the question of whether the target is legitimately identified. Working in the Bayesian paradigm, we consider the grey zone of situations where a key assumption, in the form of a parameter space restriction, is scientifically reasonable but not incontrovertible for the problem being tackled. Specifically, we investigate statistical properties that ensue if we structure a prior distribution to assert that maybe or perhaps the assumption holds. Technically this simply devolves to using a mixture prior distribution putting just some prior weight on the assumption, or one of several assumptions, holding. However, while the construct is straightforward, there is very little literature discussing situations where Bayesian model averaging is employed across a mix of fully identified and partially identified models.

Bayesian analyses combine information represented by different terms in a joint Bayesian model. When one or more of the terms is misspecified, it can be helpful to restrict the use of information from suspect model components to modify posterior inference. This is called “cutting feedback”, and both the specification and computation of the posterior for such “cut models” is challenging. In this paper, we define cut posterior distributions as solutions to constrained optimization problems, and propose variational methods for their computation. These methods are faster than existing Markov chain Monte Carlo (MCMC) approaches by an order of magnitude. It is also shown that variational methods allow for the evaluation of computationally intensive conflict checks that can be used to decide whether or not feedback should be cut. Our methods are illustrated in a number of simulated and real examples, including an application where recent methodological advances that combine variational inference and MCMC within the variational optimization are used.

In the first published treatment of the least squares, Legendre applied his new method to the French meridian data measured for the determination of the length of the meter (Nouvelles méthodes pour la détermination des orbites des comètes (1805) 76 Firmin Didot). Legendre treated one error term as a constant. It is shown here that it turns out to be equivalent to the generalized least squares solution of his model (Nouvelles méthodes pour la détermination des orbites des comètes (1805) 77 Firmin Didot).

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.