Continuously indexed datasets with multiple variables have become ubiquitous in the geophysical, ecological, environmental and climate sciences, and pose substantial analysis challenges to scientists and statisticians. For many years, scientists developed models that aimed at capturing the spatial behavior for an individual process; only within the last few decades has it become commonplace to model multiple processes jointly. The key difficulty is in specifying the cross-covariance function, that is, the function responsible for the relationship between distinct variables. Indeed, these cross-covariance functions must be chosen to be consistent with marginal covariance functions in such a way that the second-order structure always yields a nonnegative definite covariance matrix. We review the main approaches to building cross-covariance models, including the linear model of coregionalization, convolution methods, the multivariate Matérn and nonstationary and space–time extensions of these among others. We additionally cover specialized constructions, including those designed for asymmetry, compact support and spherical domains, with a review of physics-constrained models. We illustrate select models on a bivariate regional climate model output example for temperature and pressure, along with a bivariate minimum and maximum temperature observational dataset; we compare models by likelihood value as well as via cross-validation co-kriging studies. The article closes with a discussion of unsolved problems.

Physical processes rarely occur in isolation, rather they influence and interact with one another. Thus, there is great benefit in modeling potential dependence between both spatial locations and different processes. It is the interaction between these two dependencies that is the focus of Genton and Kleiber’s paper under discussion. We see the problem of ensuring that any multivariate spatial covariance matrix is nonnegative definite as important, but we also see it as a means to an end. That “end” is solving the scientific problem of predicting a multivariate field.

Although cokriging in theory should yield smaller or equal prediction variance than kriging, this outperformance sometimes is hard to see in practice. This should motivate theoretical studies on cokriging. In general, there is a lack of theoretical results for cokriging. In this work, we provide some theoretical results to compare cokriging with kriging by examining some explicit models and specific sampling schemes.

The modeling and analysis of networks and network data has seen an explosion of interest in recent years and represents an exciting direction for potential growth in statistics. Despite the already substantial amount of work done in this area to date by researchers from various disciplines, however, there remain many questions of a decidedly foundational nature—natural analogues of standard questions already posed and addressed in more classical areas of statistics—that have yet to even be posed, much less addressed. Here we raise and consider one such question in connection with network modeling. Specifically, we ask, “Given an observed network, what is the sample size?” Using simple, illustrative examples from the class of exponential random graph models, we show that the answer to this question can very much depend on basic properties of the networks expected under the model, as the number of vertices $n_{V}$ in the network grows. In particular, adopting the (asymptotic) scaling of the variance of the maximum likelihood parameter estimates as a notion of effective sample size ($n_{\mathrm{eff}}$), we show that when modeling the overall propensity to have ties and the propensity to reciprocate ties, whether the networks are sparse or not under the model (i.e., having a constant or an increasing number of ties per vertex, respectively) is sufficient to yield an order of magnitude difference in $n_{\mathrm{eff}}$, from $O(n_{V})$ to $O(n^{2}_{V})$. In addition, we report simulation study results that suggest similar properties for models for triadic (friend-of-a-friend) effects. We then explore some practical implications of this result, using both simulation and data on food-sharing from Lamalera, Indonesia.

Multi-armed bandit problems (MABPs) are a special type of optimal control problem well suited to model resource allocation under uncertainty in a wide variety of contexts. Since the first publication of the optimal solution of the classic MABP by a dynamic index rule, the bandit literature quickly diversified and emerged as an active research topic. Across this literature, the use of bandit models to optimally design clinical trials became a typical motivating application, yet little of the resulting theory has ever been used in the actual design and analysis of clinical trials. To this end, we review two MABP decision-theoretic approaches to the optimal allocation of treatments in a clinical trial: the infinite-horizon Bayesian Bernoulli MABP and the finite-horizon variant. These models possess distinct theoretical properties and lead to separate allocation rules in a clinical trial design context. We evaluate their performance compared to other allocation rules, including fixed randomization. Our results indicate that bandit approaches offer significant advantages, in terms of assigning more patients to better treatments, and severe limitations, in terms of their resulting statistical power. We propose a novel bandit-based patient allocation rule that overcomes the issue of low power, thus removing a potential barrier for their use in practice.

We compare several confidence intervals after model selection in the setting recently studied by Berk et al. [ Ann. Statist. 41 (2013) 802–837], where the goal is to cover not the true parameter but a certain nonstandard quantity of interest that depends on the selected model. In particular, we compare the PoSI-intervals that are proposed in that reference with the “naive” confidence interval, which is constructed as if the selected model were correct and fixed a priori (thus ignoring the presence of model selection). Overall, we find that the actual coverage probabilities of all these intervals deviate only moderately from the desired nominal coverage probability. This finding is in stark contrast to several papers in the existing literature, where the goal is to cover the true parameter.

Most of the consistency analyses of Bayesian procedures for variable selection in regression refer to pairwise consistency, that is, consistency of Bayes factors. However, variable selection in regression is carried out in a given class of regression models where a natural variable selector is the posterior probability of the models.

In this paper we analyze the consistency of the posterior model probabilities when the number of potential regressors grows as the sample size grows. The novelty in the posterior model consistency is that it depends not only on the priors for the model parameters through the Bayes factor, but also on the model priors, so that it is a useful tool for choosing priors for both models and model parameters.

We have found that some classes of priors typically used in variable selection yield posterior model inconsistency, while mixtures of these priors improve this undesirable behavior.

For moderate sample sizes, we evaluate Bayesian pairwise variable selection procedures by comparing their frequentist Type I and II error probabilities. This provides valuable information to discriminate between the priors for the model parameters commonly used for variable selection.

Bayesian model selection poses two main challenges: the specification of parameter priors for all models, and the computation of the resulting Bayes factors between models. There is now a large literature on automatic and objective parameter priors in the linear model. One important class are $g$-priors, which were recently extended from linear to generalized linear models (GLMs). We show that the resulting Bayes factors can be approximated by test-based Bayes factors (Johnson [ Scand. J. Stat. 35 (2008) 354–368]) using the deviance statistics of the models. To estimate the hyperparameter $g$, we propose empirical and fully Bayes approaches and link the former to minimum Bayes factors and shrinkage estimates from the literature. Furthermore, we describe how to approximate the corresponding posterior distribution of the regression coefficients based on the standard GLM output. We illustrate the approach with the development of a clinical prediction model for 30-day survival in the GUSTO-I trial using logistic regression.

Robert C. Elston was born on February 4, 1932, in London, England. He went to Cambridge University to study natural science from 1952–1956 and obtained B.A., M.A. and Diploma in Agriculture (Dip Ag). He came to the US at age 24 to study animal breeding at Cornell University and received his Ph.D. in 1959. From 1959–1960, he was a post-doctoral fellow in biostatistics at University of North Carolina (UNC), Chapel Hill, where he studied mathematical statistics. He then rose through the academic ranks in the department of biostatistics at UNC, becoming a full professor in 1969. From 1979–1995, he was a professor and head of the Department of Biometry and Genetics at Louisiana State University Medical Center in New Orleans. In 1995, he moved to Case Western Reserve University where he is a professor of epidemiology and biostatistics and served as chairman from 2008 to 2014. Between 1966 and 2013, he directed 42 Ph.D. students and mentored over 40 post-doctoral fellows. If one regards him as a founder of a pedigree in research in genetic epidemiology, it was estimated in 2007 that there were more than 500 progeny. Among his many honors are a NIH Research Career Development Award (1966–1976), the Leadership Award from International Society of Human Genetics (1995), William Allan Award from American Society of Human Genetics (1996), NIH MERIT Award (1998) and the Marvin Zelen Leadership Award, Harvard University (2004). He is a Fellow of the American Statistical Association and the Institute of Mathematical Statistics as well as a Fellow of the Ohio Academy of Science. A leader in research in genetic epidemiology for over 40 years, he has published over 600 research articles in biostatistics, genetic epidemiology and applications. He has also coauthored and edited 9 books in biostatistics, population genetics and methods for the analysis of genetic data.

The original conversation took place on August 4, 2009, during the Joint Statistical Meetings (JSM) in Washington, DC by GZ and ZL. NLG had dinner with RCE during the 2013 JSM in Montreal, Canada, and added supplementary material and edited the conversation. RCE updated and clarified certain points.

Jerome H. Friedman was born in Yreka, California, USA, on December 29, 1939. He received his high school education at Yreka High School, then spent two years at Chico State College before transferring to the University of California at Berkeley in 1959. He completed an undergraduate degree in physics in 1962 and a Ph.D. in high-energy particle physics in 1968 and was a post-doctoral research physicist at the Lawrence Berkeley Laboratory during 1968–1972. In 1972, he moved to Stanford Linear Accelerator Center (SLAC) as head of the Computation Research Group, retaining this position until 2006. In 1981, he was appointed half time as Professor in the Department of Statistics, Stanford University, remaining half time with his SLAC appointment. He has held visiting appointments at CSIRO in Sydney, CERN and the Department of Statistics at Berkeley, and has had a very active career as a commercial consultant. Jerry became Professor Emeritus in the Department of Statistics in 2007. Apart from some 30 publications in high-energy physics early in his career, Jerry has published over 70 research articles and books in statistics and computer science, including co-authoring the pioneering books Classification and Regression Trees and The Elements of Statistical Learning. Many of his publications have hundreds if not thousands of citations (e.g., the CART book has over 21,000). Much of his software is incorporated in commercial products, including at least one popular search engine. Many of his methods and algorithms are essential inclusions in modern statistical and data mining packages. Honors include the following: the Rietz Lecture (1999) and the Wald Lectures (2009); election to the American Academy of Arts and Sciences (2005) and the US National Academy of Sciences (2010); a Fellow of the American Statistical Association; Paper of the Year ( JASA 1980, 1985; Technometrics 1998, 1992); Statistician of the Year (ASA, Chicago Chapter, 1999); ACM Data Mining Lifetime Innovation Award (2002), Emanuel & Carol Parzen Award for Statistical Innovation (2004); Noether Senior Lecturer (American Statistical Association, 2010); and the IEEE Computer Society Data Mining Research Contribution Award (2012).

The interview was recorded at his home in Palo Alto, California during 3–4 August 2012.