Poisson Dirichlet Research Articles

A New Probability Measure-Valued Stochastic Process with Ferguson-Dirichlet Process as Reversible Measure

A new diffusion process taking values in the space of all probability measures over $[0,1]$ is constructed through Dirichlet form theory in this paper. This process is reversible with respect to the Ferguson-Dirichlet process (also called Poisson Dirichlet process), which is the reversible measure of the Fleming-Viot process with parent independent mutation. The intrinsic distance of this process is in the class of Wasserstein distances, so it's also a kind of Wasserstein diffusion. Moreover, this process satisfies the Log-Sobolev inequality.

Vectors of two-parameter Poisson–Dirichlet processes

The definition of vectors of dependent random probability measures is a topic of interest in applications to Bayesian statistics. They represent dependent nonparametric prior distributions that are useful for modelling observables for which specific covariate values are known. In this paper we propose a vector of two-parameter Poisson–Dirichlet processes. It is well-known that each component can be obtained by resorting to a change of measure of a σ -stable process. Thus dependence is achieved by applying a Lévy copula to the marginal intensities. In a two-sample problem, we determine the corresponding partition probability function which turns out to be partially exchangeable. Moreover, we evaluate predictive and posterior distributions.

Asymptotic results for coalescent processes without proper frequencies and applications to the two-parameter Poisson–Dirichlet coalescent

The class of coalescent processes with simultaneous multiple collisions ( Ξ -coalescents) without proper frequencies is considered. We study the asymptotic behavior of the external branch length, the total branch length and the number of mutations on the genealogical tree as the sample size n tends to infinity. The limiting random variables arising are characterized via exponential integrals of the subordinator associated with the frequency of singletons of the coalescent. The proofs are based on decompositions into external and internal branches. The asymptotics of the external branches is treated via the method of moments. The internal branches do not contribute to the limiting variables since the number C n of collisions for coalescents without proper frequencies is asymptotically negligible compared to n . The results are applied to the two-parameter Poisson–Dirichlet coalescent indicating that this particular class of coalescent processes in many respects behaves approximately as the star-shaped coalescent.

Asymptotic results for the two-parameter Poisson–Dirichlet distribution

The two-parameter Poisson–Dirichlet distribution is the law of a sequence of decreasing nonnegative random variables with total sum one. It can be constructed from stable and gamma subordinators with the two parameters, α and θ , corresponding to the stable component and the gamma component respectively. The moderate deviation principle is established for the distribution when θ approaches infinity, and the large deviation principle is established when both α and θ approach zero.

Bayesian Non-Parametric Inference for Species Variety with a Two-Parameter Poisson–Dirichlet Process Prior

SummaryA Bayesian non-parametric methodology has been recently proposed to deal with the issue of prediction within species sampling problems. Such problems concern the evaluation, conditional on a sample of size n, of the species variety featured by an additional sample of size m. Genomic applications pose the additional challenge of having to deal with large values of both n and m. In such a case the computation of the Bayesian non-parametric estimators is cumbersome and prevents their implementation. We focus on the two-parameter Poisson–Dirichlet model and provide completely explicit expressions for the corresponding estimators, which can be easily evaluated for any sizes of n and m. We also study the asymptotic behaviour of the number of new species conditionally on the observed sample: such an asymptotic result, combined with a suitable simulation scheme, allows us to derive asymptotic highest posterior density intervals for the estimates of interest. Finally, we illustrate the implementation of the proposed methodology by the analysis of five expressed sequence tags data sets.

Regenerative tree growth: Binary self-similar continuum random trees and Poisson–Dirichlet compositions

We use a natural ordered extension of the Chinese Restaurant Process to grow a twoparameter family of binary self-similar continuum fragmentation trees. We provide an explicit embedding of Ford’s sequence of alpha model trees in the continuum tree which we identified in a previous article as a distributional scaling limit of Ford’s trees. In general, the Markov branching trees induced by the two-parameter growth rule are not sampling consistent, so the existence of compact limiting trees cannot be deduced from previous work on the sampling consistent case. We develop here a new approach to establish such limits, based on regenerative interval partitions and the urn-model description of sampling from Dirichlet random distributions. AMS 2000 subject classifications: 60J80.

Some diffusion processes associated with two parameter Poisson–Dirichlet distribution and Dirichlet process

The two parameter Poisson–Dirichlet distribution PD(α, θ) is the distribution of an infinite dimensional random discrete probability. It is a generalization of Kingman’s Poisson–Dirichlet distribution. The two parameter Dirichlet process \({\Pi_{\alpha,\theta,\nu_0}}\) is the law of a pure atomic random measure with masses following the two parameter Poisson–Dirichlet distribution. In this article we focus on the construction and the properties of the infinite dimensional symmetric diffusion processes with respective symmetric measures PD(α, θ) and \({\Pi_{\alpha,\theta,\nu_0}}\). The methods used come from the theory of Dirichlet forms.

A Central Limit Theorem Associated with the Transformed Two-Parameter Poisson–Dirichlet Distribution

In this paper we introduce the transformed two-parameter Poisson–Dirichlet distributionon the ordered infinite simplex. Furthermore, we prove the central limit theorem related to this distribution when both the mutation rateθand the selection rateσbecome large in a specified manner. As a consequence, we find that the properly scaled homozygosities have asymptotical normal behavior. In particular, there is a certain phase transition with the limit depending on the relative strength ofσandθ.

A Central Limit Theorem Associated with the Transformed Two-Parameter Poisson–Dirichlet Distribution

In this paper we introduce the transformed two-parameter Poisson–Dirichlet distribution on the ordered infinite simplex. Furthermore, we prove the central limit theorem related to this distribution when both the mutation rate θ and the selection rate σ become large in a specified manner. As a consequence, we find that the properly scaled homozygosities have asymptotical normal behavior. In particular, there is a certain phase transition with the limit depending on the relative strength of σ and θ.

Sampling from Dirichlet partitions: estimating the number of species

AbstractThe Dirichlet partition of an interval can be viewed as the generalization of several classical models in ecological statistics. We recall the unordered Ewens sampling formulae ‐ESF) from finite Dirichlet partitions. As this is a key variable for estimation purposes, focus is on the number of distinct visited species in the sampling process. These are illustrated in specific cases. We use these preliminary statistical results on frequencies distribution to address the following sampling problem: what is the estimated number of species when sampling is from Dirichlet populations? The obtained results are in accordance with the ones found in sampling theory from random proportions with Poisson–Dirichlet ‐PD) distribution. To conclude with, we apply the different estimators suggested to two different sets of real data. Copyright © 2009 John Wiley & Sons, Ltd.

A phase transition behavior for Brownian motions interacting through their ranks

Consider a time-varying collection of n points on the positive real axis, modeled as Exponentials of n Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. If at each time point we divide the points by their sum, under suitable assumptions the rescaled point process converges to a stationary distribution (depending on n and the vector of drifts) as time goes to infinity. This stationary distribution can be exactly computed using a recent result of Pal and Pitman. The model and the rescaled point process are both central objects of study in models of equity markets introduced by Banner, Fernholz, and Karatzas. In this paper, we look at the behavior of this point process under the stationary measure as n tends to infinity. Under a certain ‘continuity at the edge’ condition on the drifts, we show that one of the following must happen: either (i) all points converge to 0, or (ii) the maximum goes to 1 and the rest go to 0, or (iii) the processes converge in law to a non-trivial Poisson–Dirichlet distribution. The underlying idea of the proof is inspired by Talagrand’s analysis of the low temperature phase of Derrida’s Random Energy Model of spin glasses. The main result establishes a universality property for the BFK models and aids in explicit asymptotic computations using known results about the Poisson–Dirichlet law.

Veneziano amplitudes, spin chains and Abelian reduction of QCD

Although QCD can be treated perturbatively in the high energy limit, lower energies require uses of nonperturbative methods such as ADS/CFT and/or Abelian reduction. These methods are not equivalent. While the first is restricted to supersymmetric Yang–Mills model with number of colors going to infinity, the second is not restricted by requirements of supersymmetry and is designed to work in the physically realistic limit of a finite number of colors. In this paper we provide arguments in favor of the Abelian reduction methods. This is achieved by further developing results of our recent works re-analyzing Veneziano and Veneziano-like amplitudes and the models associated with these amplitudes. It is shown, that the obtained new partition function for these amplitudes can be mapped exactly into that for the Polychronakos–Frahm (P–F) spin chain model recoverable from the Richardon–Gaudin (R–G) XXX spin chain model originally designed for treatments of the BCS-type superconductivity. Because of this, it is demonstrated that the obtained mapping is compatible with the method of Abelian reduction. The R–G model is recovered from the asymptotic (WKB-type) solutions of the rational Knizhnik–Zamolodchikov (K–Z) equation. Linear independence of these solutions is controlled by determinants whose explicit form (up to a constant) coincides with Veneziano (or Veneziano-like) amplitudes. In the simplest case, the determinantal conditions coincide with those discovered by Kummer in the 19th century. Kummer’s results admit physical interpretation by relating determinantal formula(s) to Veneziano-like amplitudes. Furthermore, these amplitudes can be interpreted as Poisson–Dirichlet distributions playing a central role in the stochastic theory of random coagulation–fragmentation processes. Such an interpretation is complementary to that known for the Lund model widely used for the description of coagulation–fragmentation processes in QCD.

Poisson–Dirichlet distribution with small mutation rate

A large deviation principle is established for the Poisson–Dirichlet distribution when the mutation rate θ converges to zero. The rate function is identified explicitly, and takes on finite values only on states that have finite number of alleles. This result is then applied to the study of the asymptotic behavior of the homozygosity, and the Poisson–Dirichlet distribution with selection. The latter shows that several alleles can coexist when selection intensity goes to infinity in a particular way as θ approaches zero.

Invariant measures for the continual Cartan subgroup

We construct and study the one-parameter semigroup of σ-finite measures L θ , θ > 0 , on the space of Schwartz distributions that have an infinite-dimensional abelian group of linear symmetries; this group is a continual analog of the classical Cartan subgroup of diagonal positive matrices of the group SL ( n , R ) . The parameter θ is the degree of homogeneity with respect to homotheties of the space, we prove uniqueness theorem for measures with given degree of homogeneity, and call the measure with degree of homogeneity equal to one the infinite-dimensional Lebesgue measure L . The structure of these measures is very closely related to the so-called Poisson–Dirichlet measures PD ( θ ) , and to the well-known gamma process. The nontrivial properties of the Lebesgue measure are related to the superstructure of the measure PD ( 1 ) , which is called the conic Poisson–Dirichlet measure— CPD. This is the most interesting σ-finite measure on the set of positive convergent monotonic real series.

Moderate deviations for Poisson–Dirichlet distribution

The Poisson--Dirichlet distribution arises in many different areas. The parameter $\theta$ in the distribution is the scaled mutation rate of a population in the context of population genetics. The limiting case of $\theta$ approaching infinity is practically motivated and has led to new, interesting mathematical structures. Laws of large numbers, fluctuation theorems and large-deviation results have been established. In this paper, moderate-deviation principles are established for the Poisson--Dirichlet distribution, the GEM distribution, the homozygosity, and the Dirichlet process when the parameter $\theta$ approaches infinity. These results, combined with earlier work, not only provide a relatively complete picture of the asymptotic behavior of the Poisson--Dirichlet distribution for large $\theta$, but also lead to a better understanding of the large deviation problem associated with the scaled homozygosity. They also reveal some new structures that are not observed in existing large-deviation results.

Two-Parameter Poisson–Dirichlet Measures and Reversible Exchangeable Fragmentation–Coalescence Processes

We show that for 0<α<1 and θ>−α, the Poisson–Dirichlet distribution with parameter (α, θ) is the unique reversible distribution of a rather natural fragmentation–coalescence process. This completes earlier results in the literature for certain split-and-merge transformations and the parameter α = 0.

Dispersion of growth paths of macroeconomic models in thermodynamic limits: two-parameter Poisson–Dirichlet models

This paper discusses dispersion of growth patterns of macroeconomic models in thermodynamic limits. More specifically, the paper shows that the coefficients of variations of the total numbers of clusters and the numbers of clusters of specific sizes of one- and two-parameter Poisson–Dirichlet models behave qualitatively differently in the thermodynamic limits. The coefficients of variations of the numbers of clusters in the former class of distributions are all self-averaging, while the those in the latter class are all non-self averaging. In other words, dispersions or variations of growth rates about the means do not vanish in the two-parameter version of the model, while they do in the one-parameter version in the thermodynamic limits. The paper ends by pointing out other models, such as triangular urn models, may converge to Mittag–Leffler distributions which exhibit non-self-averaging behavior for certain parameter combinations.

Review on solving the forward problem in EEG source analysis

BackgroundThe aim of electroencephalogram (EEG) source localization is to find the brain areas responsible for EEG waves of interest. It consists of solving forward and inverse problems. The forward problem is solved by starting from a given electrical source and calculating the potentials at the electrodes. These evaluations are necessary to solve the inverse problem which is defined as finding brain sources which are responsible for the measured potentials at the EEG electrodes.MethodsWhile other reviews give an extensive summary of the both forward and inverse problem, this review article focuses on different aspects of solving the forward problem and it is intended for newcomers in this research field.ResultsIt starts with focusing on the generators of the EEG: the post-synaptic potentials in the apical dendrites of pyramidal neurons. These cells generate an extracellular current which can be modeled by Poisson's differential equation, and Neumann and Dirichlet boundary conditions. The compartments in which these currents flow can be anisotropic (e.g. skull and white matter). In a three-shell spherical head model an analytical expression exists to solve the forward problem. During the last two decades researchers have tried to solve Poisson's equation in a realistically shaped head model obtained from 3D medical images, which requires numerical methods. The following methods are compared with each other: the boundary element method (BEM), the finite element method (FEM) and the finite difference method (FDM). In the last two methods anisotropic conducting compartments can conveniently be introduced. Then the focus will be set on the use of reciprocity in EEG source localization. It is introduced to speed up the forward calculations which are here performed for each electrode position rather than for each dipole position. Solving Poisson's equation utilizing FEM and FDM corresponds to solving a large sparse linear system. Iterative methods are required to solve these sparse linear systems. The following iterative methods are discussed: successive over-relaxation, conjugate gradients method and algebraic multigrid method.ConclusionSolving the forward problem has been well documented in the past decades. In the past simplified spherical head models are used, whereas nowadays a combination of imaging modalities are used to accurately describe the geometry of the head model. Efforts have been done on realistically describing the shape of the head model, as well as the heterogenity of the tissue types and realistically determining the conductivity. However, the determination and validation of the in vivo conductivity values is still an important topic in this field. In addition, more studies have to be done on the influence of all the parameters of the head model and of the numerical techniques on the solution of the forward problem.

On one property of Derrida–Ruelle cascades

We present new results about Poisson–Dirichlet point processes and Derrida–Ruelle cascades that are motivated by applications to mean-field spin glass models and, in particular, by the attempt to express Guerra's interpolation in the Sherrington–Kirkpatrick model entirely in the language of the cascades. To cite this article: D. Panchenko, M. Talagrand, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Large deviations associated with Poisson–Dirichlet distribution and Ewens sampling formula

Several results of large deviations are obtained for distributions that are associated with the Poisson–Dirichlet distribution and the Ewens sampling formula when the parameter θ approaches infinity. The motivation for these results comes from a desire of understanding the exact meaning of θ going to infinity. In terms of the law of large numbers and the central limit theorem, the limiting procedure of θ going to infinity in a Poisson–Dirichlet distribution corresponds to a finite allele model where the mutation rate per individual is fixed and the number of alleles going to infinity. We call this the finite allele approximation. The first main result of this article is concerned with the relation between this finite allele approximation and the Poisson–Dirichlet distribution in terms of large deviations. Large θ can also be viewed as a limiting procedure of the effective population size going to infinity. In the second result a comparison is done between the sample size and the effective population size based on the Ewens sampling formula.