超越中餐馆过程和Pitman-Yor过程：具有双幂律行为的统计模型
Beyond the Chinese Restaurant and Pitman-Yor processes: Statistical Models with Double Power-law Behavior
AI总结：
本文提出一类具有双重正则变化的完全随机测度，使归一化后的随机划分呈现双幂律行为，并给出三个具体模型及高效MCMC估计算法，在多个数据集上优于Pitman-Yor过程。
摘要：
贝叶斯非参数方法，特别是Pitman-Yor过程及其相关的双参数中餐馆过程，已成功应用于数据呈现幂律行为的场景，例如自然语言处理、自然图像和网络。越来越多的经验证据表明，一些数据集呈现出双区域幂律行为：一个区域对应低频，另一个区域以不同的指数对应高频。在本文中，我们引入一类具有双重正则变化的完全随机测度。与Pitman-Yor过程不同，我们证明当该类中的完全随机测度被归一化以获得随机概率测度及相关的随机划分时，这些划分会表现出双幂律行为。我们特别讨论了该类中的三个模型：beta prime过程（Broderick等人，2015, 2018）、一种称为广义BFRY过程的新过程，以及一种混合构造。我们推导了高效的马尔可夫链蒙特卡洛算法来估计这些模型的参数。最后，我们表明所提出的模型在多个数据集上比Pitman-Yor过程提供了更好的拟合。
英文摘要：
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.
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