Exchangeable and partially exchangeable random partitions

@article{Pitman1995ExchangeableAP,
  title={Exchangeable and partially exchangeable random partitions},
  author={Jim Pitman},
  journal={Probability Theory and Related Fields},
  year={1995},
  volume={102},
  pages={145-158}
}
  • J. Pitman
  • Published 1 June 1995
  • Mathematics
  • Probability Theory and Related Fields
SummaryCall a random partition of the positive integerspartially exchangeable if for each finite sequence of positive integersn1,...,nk, the probability that the partition breaks the firstn1+...+nk integers intok particular classes, of sizesn1,...,nk in order of their first elements, has the same valuep(n1,...,nk) for every possible choice of classes subject to the sizes constraint. A random partition is exchangeable iff it is partially exchangeable for a symmetric functionp(n1,...nk). A… 
View on Springer
stat.berkeley.edu
526 Citations
Highly Influential Citations
102
Background Citations
250
Methods Citations
171
Results Citations
12

526 Citations

Citation Type

Citation Type

Characterizations of exchangeable partitions and random discrete distributions by deletion properties
We prove a long-standing conjecture which characterises the Ewens-Pitman two- parameter family of exchangeable random partitions, plus a short list of limit and exceptional cases, by the following…
The distribution of the number of distinct values in a finite exchangeable sequence
Let Kn denote the number of distinct values among the first n terms of an infinite exchangeable sequence of random variables (X1, X2, . . .). We prove for n = 3 that the extreme points of the convex…
Extreme sizes in Gibbs-type exchangeable random partitions
Gibbs-type exchangeable random partition, which is a class of multiplicative measures on the set of positive integer partitions, appear in various contexts, including Bayesian statistics, random…
Coherent random allocations, and the Ewens-Pitman formula
Assume that there is a random number K of positive integer random variables S1, …, SK that are conditionally independent given K and all have identical distributions. A random integer partition N =…
Binary sequential representations of random partitions
TLDR
A two-parameter generalization of Ewens' partition is obtained by considering random partitions constructed from discrete renewal processes and introducing a convolution-type product on 0-1 sequences.
Increments of Random Partitions
For any partition of {1, 2,. .. , n} we define its increments Xi, 1 ≤ i ≤ n by Xi = 1 if i is the smallest element in the partition block that contains it, Xi = 0 otherwise. We prove that for…
Increments of Random Partitions
  • Serban Nacu
  • Mathematics, Computer Science
    Combinatorics, Probability and Computing
  • 2006
TLDR
The Chinese Restaurant Process CRP (the partition with distribution given by the Ewens sampling formula with parameter $\theta$) is the only exchangeable random partition with independent increments because of the law of the increments.
Frequency of Frequencies Distributions and Size-Dependent Exchangeable Random Partitions
ABSTRACT Motivated by the fundamental problem of modeling the frequency of frequencies (FoF) distribution, this article introduces the concept of a cluster structure to define a probability function…
Ordered sizes in exchangeable random partitions and their asymptotics
Some distributional results on ordered sizes in exchangeable random partitions of a natural number are obtained by combinatorial arguments. Analysis of generating functions yields their asymptotics.…
Record Indices and Age-Ordered Frequencies in Exchangeable Gibbs Partitions
The frequencies of an exchangeable Gibbs random partition of the integers (Gnedin and Pitman 2005) are considered in their age-order, i.e. their size-biased order. We study their dependence on the…
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 34 REFERENCES
Continuity and weak convergence of ranked and size-biased permutations on the infinite simplex
Partition structures derived from Brownian motion and stable subordinators
Explicit formulae are obtained for the distribution of various random partitions of a positive integer n, both ordered and unordered, derived from the zero set M of a Brownian motion by the following…
The heaps process, libraries, and size-biased permutations
The heaps process (also known as a Tsetlin library) provides a model for a self-regulating filing system. Items are requested from time to time according to their popularity and returned to the top…
The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator
The two-parameter Poisson-Dirichlet distribution, denoted PD(α,θ), is a probability distribution on the set of decreasing positive sequences with sum 1. The usual Poisson-Dirichlet distribution with…
Partition structures, Polya urns, the Ewens sampling formula, and the ages of alleles.
  • P. Donnelly
  • Mathematics
    Theoretical population biology
  • 1986
Size-biased sampling of Poisson point processes and excursions
SummarySome 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.…
Ferguson Distributions Via Polya Urn Schemes
Let p be any finite positive measure on (the Borel sets of) a complete separable metric space X. We shall say that a random probability measure P* on X has a Ferguson distribution with parameter p if…
Random discrete distributions invariant under size-biased permutation
  • J. Pitman
  • Mathematics
    Advances in Applied Probability
  • 1996
Invariance of a random discrete distribution under size-biased permutation is equivalent to a conjunction of symmetry conditions on its finite-dimensional distributions. This is applied to…
The stationary distribution of the infinitely-many neutral alleles diffusion model
An expression is found for the stationary density of the allele frequencies, in the infinitely-many alleles model. It is assumed that all alleles are neutral, that there is a constant mutation rate,…
Random partitions in population genetics
  • J. Kingman
  • Biology
    Proceedings of the Royal Society of London. Series B. Biological Sciences
  • 1978
TLDR
The sampling formula asserts that, if a sample of n gametes is taken from a population, and classified according to the gene at a particular locus, then the probability that there are a1 alleles represented once in the sample, a2 represented twice, a3 thrice, and so on, is given for some positive value of θ.
...
1
2
3
4
...