Bayesian Nonparametric Modeling and the Ubiquitous Ewens Sampling Formula
Yee Whye Teh · Statistical Science · 2016
I would like to thank Harry Crane for a most enlightening review of the many ways and guises in which the Ewens sampling formula pops up throughout statistics and mathematics.Given the simplicity and the almost inevitability of Ewens' sampling formula when working with distributions over partitions, one could say that it plays a similar role for random partitions as the normal distribution plays for random real-valued variables.And just as the normal distribution plays an important role as a core building block for more complex models, for example, hierarchical Bayesian models or graphical models, Ewens' sampling formula and the associated Chinese restaurant process distribution over set partitions and Dirichlet process distribution over probability measures play increasingly important roles as building blocks of more complex Bayesian nonparametric models.Crane has noted, and I agree, that this is "one of the most active areas of statistical research," whose "overwhelming activity forbids any possibility of a satisfactory survey of the topic and promises to quickly outdate the contents of the present section."In this discussion I will attempt to present an (already outdated) overview of the use of Ewens' sampling formula in Bayesian nonparametrics, specifically focusing on the many creative ways the community has built more complex models out of these simpler building blocks.Much of the work is motivated by recent trends toward using the analysis of "Big Data" sets to derive scientific understanding and drive technological progress.Such modern data sets are often not just tall, they are also wide, and not just tall and wide, but also complex and structured, and it is important to model the nontrivial dependencies hidden behind the data.Good introductions to Bayesian nonparametrics can be found in the collection edited by Hjort et al.