Stick-breaking processes with exchangeable length variables

Spatio-temporal stick-breaking process

Dirichlet processes and their extensions have reached a great popularity in Bayesian nonparametric statistics. They have also been introduced for spatial and spatio-temporal data, as a tool to analyze and predict surfaces. A popular approach to Dirichlet processes in a spatial setting relies on a stick-breaking representation of the process, where the dependence over space is described in the definition of the stick-breaking probabilities. Extensions to include temporal dependence are still limited, however it is important, in particular for those phenomena which may change rapidly over time a…

Stick-breaking processes, clumping, and Markov chain occupation laws

We consider the connections among `clumped' residual allocation models (RAMs), a general class of stick-breaking processes including Dirichlet processes, and the occupation laws of certain discrete space time-inhomogeneous Markov chains related to simulated annealing and other applications. An intermediate structure is introduced in a given RAM, where proportions between successive indices in a list are added or clumped together to form another RAM. In particular, when the initial RAM is a Griffiths-Engen-McCloskey (GEM) sequence and the indices are given by the random times that an auxiliary …

马尔可夫碎裂过程

Markov Stick-breaking Processes

本文提出马尔可夫碎裂过程，探讨其性质和应用，揭示其在贝叶斯非参数模型中的重要性。

On the use of Markovian stick-breaking priors

In [10], a `Markovian stick-breaking' process which generalizes the Dirichlet process $(μ, θ)$ with respect to a discrete base space ${\mathfrak X}$ was introduced. In particular, a sample from from the `Markovian stick-breaking' processs may be represented in stick-breaking form $\sum_{i\geq 1} P_i δ_{T_i}$ where $\{T_i\}$ is a stationary, irreducible Markov chain on ${\mathfrak X}$ with stationary distribution $μ$, instead of i.i.d. $\{T_i\}$ each distributed as $μ$ as in the Dirichlet case, and $\{P_i\}$ is a GEM$(θ)$ residual allocation sequence. Although the motivation in [10] was to rela…

关于归一化逆高斯过程两个被忽视的棍子断裂构造

On two overlooked stick-breaking constructions of the normalized inverse Gaussian process

本文揭示了归一化逆高斯随机离散分布的两种替代棍子断裂构造，分别基于条件布朗运动划分和归一化广义Gamma子序的随机时空变换，并推广到更广泛的分布族。

A simple proof of Pitman-Yor's Chinese restaurant process from its stick-breaking representation

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 control of the clustering behaviour. Two commonly used representations of the Pitman--Yor process are the stick-breaking process and the Chinese restaurant process. The former is a constructive representation of the process which turns out very handy for practical implementation, while the latter describes the partition distribution induced. However, the usual pro…

The Stick-Breaking Construction of the Beta Process as a Poisson Process

We show that the stick-breaking construction of the beta process due to Paisley, et al. (2010) can be obtained from the characterization of the beta process as a Poisson process. Specifically, we show that the mean measure of the underlying Poisson process is equal to that of the beta process. We use this underlying representation to derive error bounds on truncated beta processes that are tighter than those in the literature. We also develop a new MCMC inference algorithm for beta processes, based in part on our new Poisson process construction.

Asymptotic behavior of the number of distinct values in a sample from the geometric stick-breaking process

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 corresponding random number of distinct values. The growth rate can be determined from the rate of decay of the small frequencies implying that, when the decreasingly ordered frequencies admit a tractable form, the asymptotics of the number of distinct values can be conveniently assessed. We focus on the geometric stick-breaking process and we investigate the effect …

基于散度的折棍过程先验分析

On a divergence-based prior analysis of stick-breaking processes

本文通过计算狄利克雷过程与几何过程之间随机KL散度的均值和方差，并将其推广到更广泛的可交换折棍过程族，建立了这些非参数先验在总变差距离下接近的定量条件，从而为按奥卡姆剃刀原则选择更简单过程提供理论依据。

Stick-breaking Pitman-Yor processes given the species sampling size

Random discrete distributions, say $F,$ known as species sampling models, represent a rich class of models for classification and clustering, in Bayesian statistics and machine learning. They also arise in various areas of probability and its applications. Jim Pitman, within the species sampling context, shows that mixed Poisson processes may be interpreted as the sample size up till a given time or in terms of waiting times of appearance of individuals to be classified. He notes connections to some recent work in the Bayesian statistic/machine learning literature, with some more classical res…