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Concentration in the Generalized Chinese Restaurant Process

Author

Listed:
  • R. I. Oliveira

    (IMPA)

  • A. Pereira

    (Universidade Federal de Alagoas)

  • R. Ribeiro

    (PUC Chile)

Abstract

The Generalized Chinese Restaurant Process (GCRP) describes a sequence of exchangeable random partitions of the numbers { 1 , … , n } $\{1,\dots ,n\}$ . 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 nα with α > 0. We prove a non-asymptotic concentration result for the number of parts of size k = o ( n α / ( 2 α + 4 ) / ( log n ) 1 / ( 2 + α ) ) $k=o(n^{\alpha /(2\alpha +4)}/(\log n)^{1/(2+\alpha )})$ . In particular, we show that these random variables concentrate around ckV∗nα where V∗nα is the asymptotic number of parts and ck ≈ 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.

Suggested Citation

  • R. I. Oliveira & A. Pereira & R. Ribeiro, 2022. "Concentration in the Generalized Chinese Restaurant Process," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(2), pages 628-670, August.
    • Handle: RePEc:spr:sankha:v:84:y:2022:i:2:d:10.1007_s13171-020-00210-7
    DOI: 10.1007/s13171-020-00210-7
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