Rejoinder: The Ubiquitous Ewens Sampling Formula

The main article and extended discussion point to Ewens’s sampling formula (ESF) as one of a few essential probability distributions. Arratia, Barbour and Tavaré explain the emergence of ESF by the Feller coupling and also touch on number theoretic considerations; Feng provides deeper background on diffusion processes and nonequilibrium versions of ESF; and McCullagh regales us with a story from the works of Fisher and Good, putting historical context around the more specialized topics covered by Favaro and James and Teh. The breadth of these comments exemplifies the expansive sphere of influence of Ewens’s sampling formula on integer partitions, Ewens’s distribution on set partitions, and the Ewens process. I thank all of the discussants for their participation in this important survey. For the most part, these contributions bolster my main thesis which, in the words of Arratia, Barbour and Tavaré, emphasizes the universal character of the Ewens sampling formula. As McCullagh notes, the contents and subsequent discussion comprise an impressive list stretching from literary studies to population genetics and probabilistic number theory. Both comments accord with my opening remark that Ewens’s sampling formula exemplifies the harmony of mathematical theory, statistical application, and scientific discovery. As a whole, however, the discussion skews disproportionately toward Bayesian nonparametrics in a way that works against the theme of ubiquity. I attempt to rebalance the conversation in these final pages.