Size-biased filtering of Poisson–Dirichlet samples with an application to partition structures in genetics

1986 ◽  
Vol 23 (4) ◽  
pp. 1008-1012 ◽  
Author(s):  
Fred M. Hoppe
Keyword(s):  
Characteristic Property ◽  
Dirichlet Distribution ◽  
Biased Sampling ◽  
Sampling Formula

A characteristic property of the Ewen sampling formula is shown to follow from the invariance under size-biased sampling of the Poisson–Dirichlet distribution.

Size-biased filtering of Poisson–Dirichlet samples with an application to partition structures in genetics

1986 ◽  
Vol 23 (04) ◽  
pp. 1008-1012
Author(s):  
Fred M. Hoppe
Keyword(s):  
Characteristic Property ◽  
Dirichlet Distribution ◽  
Biased Sampling ◽  
Sampling Formula

A characteristic property of the Ewen sampling formula is shown to follow from the invariance under size-biased sampling of the Poisson–Dirichlet distribution.

Size-biased filtering of Poisson–Dirichlet samples with an application to partition structures in genetics

1986 ◽  
Vol 23 (04) ◽  
pp. 1008-1012 ◽  
Author(s):  
Fred M. Hoppe
Keyword(s):  
Characteristic Property ◽  
Dirichlet Distribution ◽  
Biased Sampling ◽  
Sampling Formula

A characteristic property of the Ewen sampling formula is shown to follow from the invariance under size-biased sampling of the Poisson–Dirichlet distribution.

scholarly journals Gibbs partitions, Riemann–Liouville fractional operators, Mittag–Leffler functions, and fragmentations derived from stable subordinators

2021 ◽  
Vol 58 (2) ◽  
pp. 314-334
Author(s):  
Man-Wai Ho ◽  
Lancelot F. James ◽  
John W. Lau
Keyword(s):  
Special Functions ◽  
Dirichlet Distribution ◽  
Random Trees ◽  
Fractional Integrals ◽  
Biased Sampling ◽  
Scaling Limits ◽  
Fractional Operators ◽  
Random Partitions ◽  
Two Parameter ◽  
Potential Use

AbstractPitman (2003), and subsequently Gnedin and Pitman (2006), showed that a large class of random partitions of the integers derived from a stable subordinator of index $\alpha\in(0,1)$ have infinite Gibbs (product) structure as a characterizing feature. The most notable case are random partitions derived from the two-parameter Poisson–Dirichlet distribution, $\textrm{PD}(\alpha,\theta)$, whose corresponding $\alpha$-diversity/local time have generalized Mittag–Leffler distributions, denoted by $\textrm{ML}(\alpha,\theta)$. Our aim in this work is to provide indications on the utility of the wider class of Gibbs partitions as it relates to a study of Riemann–Liouville fractional integrals and size-biased sampling, and in decompositions of special functions, and its potential use in the understanding of various constructions of more exotic processes. We provide characterizations of general laws associated with nested families of $\textrm{PD}(\alpha,\theta)$ mass partitions that are constructed from fragmentation operations described in Dong et al. (2014). These operations are known to be related in distribution to various constructions of discrete random trees/graphs in [n], and their scaling limits. A centerpiece of our work is results related to Mittag–Leffler functions, which play a key role in fractional calculus and are otherwise Laplace transforms of the $\textrm{ML}(\alpha,\theta)$ variables. Notably, this leads to an interpretation within the context of $\textrm{PD}(\alpha,\theta)$ laws conditioned on Poisson point process counts over intervals of scaled lengths of the $\alpha$-diversity.

scholarly journals The sampling formula and Laplace transform associated with the two-parameter Poisson-Dirichlet distribution

2011 ◽  
Vol 43 (04) ◽  
pp. 1066-1085
Author(s):  
Fang Xu
Keyword(s):  
Laplace Transform ◽  
Random Sampling ◽  
Dirichlet Distribution ◽  
Probability Generating Function ◽  
Fixed Time ◽  
Infinite Dimensional ◽  
Sampling Formula ◽  
Dimensional Distribution ◽  
The Relationship ◽  
Two Parameter

In this paper we investigate the relationship between the sampling formula and Laplace transform associated with the two-parameter Poisson-Dirichlet distribution. We conclude that they are equivalent to determining the corresponding infinite-dimensional distribution. With these tools, a central limit theorem is established associated with the infinitely-many-neutral-alleles model at any fixed time. We also obtain the probability generating function of random sampling from a generalized two-parameter diffusion process. At the end of the paper a selection case is considered.

Random partitions in population genetics

1978 ◽  
Vol 361 (1704) ◽  
pp. 1-20 ◽  
Keyword(s):  
Population Genetics ◽  
Genetic Variation ◽  
Charge State ◽  
Characteristic Property ◽  
Large Population ◽  
Recurrent Mutation ◽  
Random Partitions ◽  
Ewens Sampling Formula ◽  
Sampling Formula ◽  
State Models

This paper is concerned with models for the genetic variation of a sample of gametes from a large population. The need for consistency between different sample sizes limits the mathematical possibilities to what are here called ‘partition structures Distinctive among them is the structure described by the Ewens sampling formula, which is shown to enjoy a characteristic property of non-interference between the different alleles. This characterization explains the robustness of the Ewens formula when neither selection nor recurrent mutation is significant, although different structures arise from selective and ‘charge-state’ models

scholarly journals Gaussian Limits Associated with the Poisson-Dirichlet Distribution and the Ewens Sampling Formula

2002 ◽  
Vol 12 (1) ◽  
pp. 101-124 ◽  
Author(s):  
Paul Joyce ◽  
Stephen M. Krone ◽  
Thomas G. Kurtz
Keyword(s):  
Dirichlet Distribution ◽  
Ewens Sampling Formula ◽  
Sampling Formula

scholarly journals Structure of the Condensed Phase in the Inclusion Process

2019 ◽  
Vol 178 (3) ◽  
pp. 682-710
Author(s):  
Watthanan Jatuviriyapornchai ◽  
Paul Chleboun ◽  
Stefan Grosskinsky
Keyword(s):  
Condensed Phase ◽  
Lattice Site ◽  
Large Deviation ◽  
Dirichlet Distribution ◽  
Occupation Number ◽  
Particle Systems ◽  
Biased Sampling ◽  
Partition Functions ◽  
Inclusion Process ◽  
Product Distributions

AbstractWe establish a complete picture of condensation in the inclusion process in the thermodynamic limit with vanishing diffusion, covering all scaling regimes of the diffusion parameter and including large deviation results for the maximum occupation number. We make use of size-biased sampling to study the structure of the condensed phase, which can extend over more than one lattice site and exhibit an interesting hierarchical structure characterized by the Poisson–Dirichlet distribution. While this approach is established in other areas including population genetics or random permutations, we show that it also provides a powerful tool to analyse homogeneous condensation in stochastic particle systems with stationary product distributions. We discuss the main mechanisms beyond inclusion processes that lead to the interesting structure of the condensed phase, and the connection to other generic particle systems. Our results are exact, and we present Monte-Carlo simulation data and recursive numerics for partition functions to illustrate the main points.

The distribution of the frequencies of age-ordered alleles in a diffusion model

1990 ◽  
Vol 22 (3) ◽  
pp. 519-532 ◽  
Author(s):  
S. N. Ethier
Keyword(s):  
Diffusion Model ◽  
Dirichlet Distribution ◽  
Random Variables ◽  
Sampling Formula

We prove that the frequencies of the oldest, second-oldest, third-oldest, … alleles in the stationary infinitely-many-neutral-alleles diffusion model are distributed as X1, (1 − X1)X2, (1 − X1)(1 − X2)X3, …, where X1, X2,X3, … are independent beta (1, θ) random variables, θ being twice the mutation intensity; that is, the frequencies of age-ordered alleles have the so-called Griffiths–Engen–McCloskey, or GEM, distribution. In fact, two proofs are given, the first involving reversibility and the size-biased Poisson–Dirichlet distribution, and the second relying on a result of Donnelly and Tavaré relating their age-ordered sampling formula to the GEM distribution.

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