Concentration in the Generalized Chinese Restaurant Process

The Generalized Chinese Restaurant Process (GCRP) describes a sequence of exchangeable random partitions of the numbers {1,… ,n} . This process is related to the Ewens sampling model in Genetics and to Bayesian nonparametric methods such as topic models. In this paper, we study the GCRP in a regime where the number of parts grows like n α with α > 0. We prove a non-asymptotic concentration result for the number of parts of size k=o(n^α /(2α +4)/(log n)^1/(2+α )) . In particular, we show that these random variables concentrate around c k V ∗ n α where V ∗ n α is the asymptotic number of parts and c k ≈ k −(1+ α ) is a positive value depending on k . We also obtain finite- n bounds for the total number of parts. Our theorems complement asymptotic statements by Pitman and more recent results on large and moderate deviations by Favaro, Feng and Gao.