2016/07/31 by Mingyuan Zhou, Stefano Favaro, Zhou, Mingyuan +4
Computer Science · Mathematics · #Algorithms and Data Compression #Bayesian Methods and Mixture Models #Binomial distribution #Counting process #Importance sampling #Joint probability distribution #Mathematics #Monte Carlo method #Multivariate random variable #Negative binomial distribution #Physics #Poisson distribution #Poisson sampling #Population #Random variable #Slice sampling #Statistical physics #Statistics #Stochastic processes and statistical mechanics #math.ST #stat.AP #stat.ME #stat.TH
paper · pdf · doi:10.48550/arxiv.1608.00264
published in arXiv (Cornell University) (Cornell University) · To appear in the Journal of the American Statistical Association (Theory and Methods). 26 pages + 17 page supplement, 19 figures. arXiv admin note: text overlap with arXiv:1410.3155
arxiv created 2016/07/31 · openalex publication_date 2016/07/31 · arxiv updated 2016/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08
Motivated by the fundamental problem of modeling the frequency of frequencies (FoF) distribution, this paper introduces the concept of a cluster structure to define a probability function that governs the joint distribution of a random count and its exchangeable random partitions. A cluster structure, naturally arising from a completely random measure mixed Poisson process, allows the probability distribution of the random partitions of a subset of a population to be dependent on the population size, a distinct and motivated feature that makes it more flexible than a partition structure. This allows it to model an entire FoF distribution whose structural properties change as the population size varies. A FoF vector can be simulated by drawing an infinite number of Poisson random variables, or by a stick-breaking construction with a finite random number of steps. A generalized negative binomial process model is proposed to generate a cluster structure, where in the prior the number of clusters is finite and Poisson distributed, and the cluster sizes follow a truncated negative binomial distribution. We propose a simple Gibbs sampling algorithm to extrapolate the FoF vector of a population given the FoF vector of a sample taken without replacement from the population. We illustrate our results and demonstrate the advantages of the proposed models through the analysis of real text, genomic, and survey data.