Related papers: Stick-breaking Pitman-Yor processes given the spec…
Stick-breaking has a long history and is one of the most popular procedures for constructing random discrete distributions in Statistics and Machine Learning. In particular, due to their intuitive construction and computational tractability…
This work centers around results related to Proposition 21 of Pitman and Yor's (1997) paper on the two parameter Poisson Dirichlet distribution indexed by (\alpha,\theta) for 0<\alpha<1, also \alpha=0, and \theta>-\alpha, denoted…
For a long time, the Dirichlet process has been the gold standard discrete random measure in Bayesian nonparametrics. The Pitman--Yor process provides a simple and mathematically tractable generalization, allowing for a very flexible…
In this article we survey properties of mixed Poisson distributions and probabilistic aspects of the Stirling transform: given a non-negative random variable $X$ with moment sequence $(\mu_s)_{s\in\mathbb{N}}$ we determine a discrete random…
The Pitman-Yor process is a random probability distribution, that can be used as a prior distribution in a nonparametric Bayesian analysis. The process is of species sampling type and generates discrete distributions, which yield of the…
The Pitman-Yor process is a random discrete measure. The random weights or masses follow the two-parameter Poisson-Dirichlet distribution with parameters $0<\alpha<1, \theta>-\alpha$. The parameters $\alpha$ and $\theta$ correspond to the…
Stacy distribution defined for the first time in 1961 provides a flexible framework for modelling of a wide range of real-life behaviours. It appears under different names in the scientific literature and contains many useful particular…
The Pitman-Yor process is a random discrete probability distribution of which the atoms can be used to model the relative abundance of species. The process is indexed by a type parameter $\sigma$, which controls the number of different…
Our object of study is the general class of stick-breaking processes with exchangeable length variables. These generalize well-known Bayesian non-parametric priors in an unexplored direction. We give conditions to assure the respective…
This paper investigates the theoretical and empirical performance of Fisher-Pitman-type permutation tests for assessing the equality of unknown Poisson mixture distributions. Building on nonparametric maximum likelihood estimators (NPMLEs)…
While most Bayesian nonparametric models in machine learning have focused on the Dirichlet process, the beta process, or their variants, the gamma process has recently emerged as a useful nonparametric prior in its own right. Current…
This paper focuses on the size-biased permutation of $n$ independent and identically distributed (i.i.d.) positive random variables. This is a finite dimensional analogue of the size-biased permutation of ranked jumps of a subordinator…
Discrete random probability measures are a key ingredient of Bayesian nonparametric inferential procedures. A sample generates ties with positive probability and a fundamental object of both theoretical and applied interest is the…
This work focuses on clustering populations with a hierarchical dependency structure that can be described by a tree. A particular example that is the focus of our work is the phylogenetic tree, with nodes often representing biological…
In this paper we consider approximations to the popular Pitman-Yor process obtained by truncating the stick-breaking representation. The truncation is determined by a random stopping rule that achieves an almost sure control on the…
In Bayesian nonparametric inference, random discrete probability measures are commonly used as priors within hierarchical mixture models for density estimation and for inference on the clustering of the data. Recently, it has been shown…
Motivated by the fundamental problem of measuring species diversity, this paper introduces the concept of a cluster structure to define an exchangeable cluster probability function that governs the joint distribution of a random count and…
We consider the sequential sampling of species, where observed samples are classified into the species they belong to. We are particularly interested in studying some quantities describing the sampling process when there is a new species…
Multitype branching processes with immigration in one type are used to model the dynamics of stage-structured plant populations. Parametric inference is first carried out when count data of all types are observed. Statistical…
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…