Ewens’s sampling formula exemplifies the harmony of mathematical theory, statistical application, and scientific discovery. The formula not only contributes to the foundations of evolutionary molecular genetics, the neutral theory of biodiversity, Bayesian nonparametrics, combinatorial stochastic processes, and inductive inference but also emerges from fundamental concepts in probability theory, algebra, and number theory. With an emphasis on its far-reaching influence throughout statistics and probability, we highlight these and many other consequences of Ewens’s seminal discovery.

Crane [Statist. Sci. 31 (2016) 1–19] provides an excellent review of Ewens’ sampling formula (henceforth, ESF), its applications in and connections with various subjects. This note intends to extend the discussion a little bit. The focus will be on nonequilibrium ESF involving diffusion processes, ESF with symmetric selection and asymptotics of ESF. The references listed are by no means exhaustive.

The mixture model devised by Fisher, Corbet and Williams [Journal of Animal Ecology 12 (1943) 42–58] for species sampling and the sequential prediction approach pioneered by Good [Biometrika 40 (1953) 237–264] and Good and Toulmin [Biometrika 43 (1956) 45–63] are both closely related to the Ewens sampling formula. Fisher’s two-parameter joint distribution for the species counts includes the Ewens distribution as the conditional distribution given the sample size. The log-series model, as it is known in the ecological literature, is closely related to a Poisson process model devised by Arratia, Barbour and Tavaré [Ann. Appl. Probab. 2 (1992) 519–535]. Oddly, despite its advantages for statistical inference, Fisher does not mention the conditional distribution. Likewise, athough Good (1953) pioneered the sequential prediction approach, neither he nor Toulmin discovered the Ewens process in a form equivalent to the modern-day Chinese restaurant process.

We commend Harry Crane on his review paper which serves to not only point out the ubiquity of the Ewens sampling formula (ESF) but also highlights some connections to more recent developments. As pointed out by Harry Crane, it is impossible to cover all aspects of the ESF and its relatives in the pages generously provided by this journal. Our task is to present additional commentary in regards to some, perhaps not so well-known, related developments in Bayesian noparametrics.

Statistical methods based on a regression model plus a zero-mean Gaussian process (GP) have been widely used for predicting the output of a deterministic computer code. There are many suggestions in the literature for how to choose the regression component and how to model the correlation structure of the GP. This article argues that comprehensive, evidence-based assessment strategies are needed when comparing such modeling options. Otherwise, one is easily misled. Applying the strategies to several computer codes shows that a regression model more complex than a constant mean either has little impact on prediction accuracy or is an impediment. The choice of correlation function has modest effect, but there is little to separate two common choices, the power exponential and the Matérn, if the latter is optimized with respect to its smoothness. The applications presented here also provide no evidence that a composite of GPs provides practical improvement in prediction accuracy. A limited comparison of Bayesian and empirical Bayes methods is similarly inconclusive. In contrast, we find that the effect of experimental design is surprisingly large, even for designs of the same type with the same theoretical properties.

The main focus of this work is on providing a formal definition of statistical depth for functional data on the basis of six properties, recognising topological features such as continuity, smoothness and contiguity. Amongst our depth defining properties is one that addresses the delicate challenge of inherent partial observability of functional data, with fulfillment giving rise to a minimal guarantee on the performance of the empirical depth beyond the idealised and practically infeasible case of full observability. As an incidental product, functional depths satisfying our definition achieve a robustness that is commonly ascribed to depth, despite the absence of a formal guarantee in the multivariate definition of depth. We demonstrate the fulfillment or otherwise of our properties for six widely used functional depth proposals, thereby providing a systematic basis for selection of a depth function.

Generalized linear mixed models are a common statistical tool for the analysis of clustered or longitudinal data where correlation is accounted for through cluster-specific random effects. In practice, the distribution of the random effects is typically taken to be a Normal distribution, although if this does not hold then the model is misspecified and standard estimation/inference may be invalid. An alternative is to perform a so-called nonparametric Bayesian analyses in which one assigns a Dirichlet process (DP) prior to the unknown distribution of the random effects. In this paper we examine operating characteristics for estimation of fixed effects and random effects based on such an analysis under a range of “true” random effects distributions. As part of this we investigate various approaches for selection of the precision parameter of the DP prior. In addition, we illustrate the use of the methods with an analysis of post-operative complications among $n=18{,}643$ female Medicare beneficiaries who underwent a hysterectomy procedure at $N=503$ hospitals in the US. Overall, we conclude that using the DP prior in modeling the random effect distribution results in large reductions of bias with little loss of efficiency. While no single choice for the precision parameter will be optimal in all settings, certain strategies such as importance sampling or empirical Bayes can be used to obtain reasonable results in a broad range of data scenarios.

Highly nonlinear, chaotic or near chaotic, dynamic models are important in fields such as ecology and epidemiology: for example, pest species and diseases often display highly nonlinear dynamics. However, such models are problematic from the point of view of statistical inference. The defining feature of chaotic and near chaotic systems is extreme sensitivity to small changes in system states and parameters, and this can interfere with inference. There are two main classes of methods for circumventing these difficulties: information reduction approaches, such as Approximate Bayesian Computation or Synthetic Likelihood, and state space methods, such as Particle Markov chain Monte Carlo, Iterated Filtering or Parameter Cascading. The purpose of this article is to compare the methods in order to reach conclusions about how to approach inference with such models in practice. We show that neither class of methods is universally superior to the other. We show that state space methods can suffer multimodality problems in settings with low process noise or model misspecification, leading to bias toward stable dynamics and high process noise. Information reduction methods avoid this problem, but, under the correct model and with sufficient process noise, state space methods lead to substantially sharper inference than information reduction methods. More practically, there are also differences in the tuning requirements of different methods. Our overall conclusion is that model development and checking should probably be performed using an information reduction method with low tuning requirements, while for final inference it is likely to be better to switch to a state space method, checking results against the information reduction approach.

Factor analysis is over a century old, but it is still problematic to choose the number of factors for a given data set. We provide a systematic review of current methods and then introduce a method based on bi-cross-validation, using randomly held-out submatrices of the data to choose the optimal number of factors. We find it performs better than many existing methods especially when both the number of variables and the sample size are large and some of the factors are relatively weak. Our performance criterion is based on recovery of an underlying signal, equal to the product of the usual factor and loading matrices. Like previous comparisons, our work is simulation based. Recent advances in random matrix theory provide principled choices for the number of factors when the noise is homoscedastic, but not for the heteroscedastic case. The simulations we chose are designed using guidance from random matrix theory. In particular, we include factors which are asymptotically too small to detect, factors large enough to detect but not large enough to improve the estimate, and two classes of factors (weak and strong) large enough to be useful. We also find that a form of early stopping regularization improves the recovery of the signal matrix.