# Beyond the Chinese Restaurant and Pitman-Yor processes: Statistical Models with Double Power-law Behavior

arXiv: https://arxiv.org/abs/1902.04714 · HTML: https://mlsanity.com/paper/1902.04714/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Feb 2019.

## Built on by (most cited first, 2 of 2)

- [Deep neural networks with dependent weights: Gaussian Process mixture limit, heavy tails, sparsity and compressibility](https://mlsanity.com/paper/2205.08187/) (2022)
- [Rapidly Varying Completely Random Measures for Modeling Extremely Sparse Networks](https://mlsanity.com/paper/2505.13206/) (2025)

## Abstract (arXiv)

Bayesian nonparametric approaches, in particular the Pitman-Yor process and the associated two-parameter Chinese Restaurant process, have been successfully used in applications where the data exhibit a power-law behavior. Examples include natural language processing, natural images or networks. There is also growing empirical evidence that some datasets exhibit a two-regime power-law behavior: one regime for small frequencies, and a second regime, with a different exponent, for high frequencies. In this paper, we introduce a class of completely random measures which are doubly regularly-varying. Contrary to the Pitman-Yor process, we show that when completely random measures in this class are normalized to obtain random probability measures and associated random partitions, such partitions exhibit a double power-law behavior. We discuss in particular three models within this class: the beta prime process (Broderick et al. (2015, 2018), a novel process called generalized BFRY process, and a mixture construction. We derive efficient Markov chain Monte Carlo algorithms to estimate the parameters of these models. Finally, we show that the proposed models provide a better fit than the Pitman-Yor process on various datasets.