Journal ArticleDOI
Size-biased sampling of Poisson point processes and excursions
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In this paper, general formulae are obtained for size-biased sampling from a Poisson point process in an abstract space where the size of a point is defined by an arbitrary strictly positive function.Abstract:
Some general formulae are obtained for size-biased sampling from a Poisson point process in an abstract space where the size of a point is defined by an arbitrary strictly positive function. These formulae explain why in certain cases (gamma and stable) the size-biased permutation of the normalized jumps of a subordinator can be represented by a stickbreaking (residual allocation) scheme defined by independent beta random variables. An application is made to length biased sampling of excursions of a Markov process away from a recurrent point of its statespace, with emphasis on the Brownian and Bessel cases when the associated inverse local time is a stable subordinator. Results in this case are linked to generalizations of the arcsine law for the fraction of time spent positive by Brownian motion.read more
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Gibbs sampling methods for stick-breaking priors
TL;DR: Two general types of Gibbs samplers that can be used to fit posteriors of Bayesian hierarchical models based on stick-breaking priors are presented and the blocked Gibbs sampler, based on an entirely different approach that works by directly sampling values from the posterior of the random measure.
BookDOI
Combinatorial Stochastic Processes
TL;DR: In this paper, the Brownian forest and the additive coalescent were constructed for random walks and random forests, respectively, and the Bessel process was used for random mappings.
Journal ArticleDOI
The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator
Jim Pitman,Marc Yor +1 more
TL;DR: The two-parameter Poisson-Dirichlet distribution with a single parameter is known as the size-biased random permutation (SBNP) as discussed by the authors, which was introduced by Engen in the context of species diversity and rediscovered by Perman and the authors in the study of excursions of Bessel processes.
Book
Logarithmic Combinatorial Structures: A Probabilistic Approach
TL;DR: In this article, the authors explain the similarities in asymptotic behaviour as the result of two basic properties shared by the structures: the conditioning relation and the logarithmic condition.
Journal ArticleDOI
Exchangeable and partially exchangeable random partitions
TL;DR: In this paper, a generalization of Ewens' partition structure, called partially exchangeable random partitions (PEBP), is presented, where a random partition of the positive integers is exchangeable iff it is partially exchangeable for a symmetric function p(n¯¯¯¯1,...,nk).
References
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Journal ArticleDOI
A Bayesian Analysis of Some Nonparametric Problems
TL;DR: In this article, a class of prior distributions, called Dirichlet process priors, is proposed for nonparametric problems, for which treatment of many non-parametric statistical problems may be carried out, yielding results that are comparable to the classical theory.
BookDOI
An introduction to the theory of point processes
Daryl J. Daley,David Vere-Jones +1 more
TL;DR: An introduction to the theory of point processes can be found in this article, where the authors introduce the concept of point process and point process theory and introduce point processes as a theory for point processes.
Journal ArticleDOI
Ferguson Distributions Via Polya Urn Schemes
TL;DR: In this article, it was shown that a random probability measure P* on X has a Ferguson distribution with parameter p if for every finite partition (B1, *. *, B) of X, the vector p*(B,), * * *, p *(B) has a Dirichlet distribution with parameters (Bj), *--, cp(B,) (when p(B), = 0, this means p*) = 0 with probability 1).
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The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator
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