arXiv subjects

Shui Feng

Publications and source records attributed to Shui Feng.

At least 19 recordsLinked to original sources

arXiv2026

Dynamical models for the two-parameter Poisson-Dirichlet distribution and the Pitman-Yor process

In this paper, we introduce and study a family of diffusion processes associated with the Pitman-Yor process and the two-parameter Poisson-Dirichlet distribution. The diffusion coefficients indexed by a non-negative parameter $\gamma$ are smaller than the corresponding one-parameter models in terms of quadratic forms or bilinear forms when $\gamma$ is positive. The well known Petrov's diffusion corresponds to $\gamma=0$ among the unlabelled diffusions. If $\gamma$ is the same as the stable parameter $\alpha$ in the Pitman-Yor process, we obtain both labelled and unlabelled reversible diffusion processes with the Pitman-Yor process and the two-parameter Poisson-Dirichlet distribution as the corresponding reversible measures. We construct these processes analytically through Dirichlet forms. In comparison with existing models in the literature, our models possess two fundamental new features. Firstly, our labelled model is characterized by an explicit generator, which is the first among all models studied so far. This makes it possible to establish the crucial integration-by-parts formula. Secondly, a novel foundational structure of the two-parameter distributions is the existence of a diversity index (a multiple of local time) for the positive stable parameter. By slowing down the diffusion in our model, the essential role of the diversity index is revealed in the evolution of the population. Additionally, we also obtain properties including ergodicity, path behaviour, and finite dimensional approximations.

math.PR↗

arXiv2026

Central limit theorem for the homozygosity of the hierarchical Pitman-Yor process

The hierarchical Pitman-Yor process is a discrete random measure used as a prior in Bayesian nonparametrics. It is motivated by the study of groups of clustered data exhibiting power law behavior. Our focus in this paper is on the Gaussian behavior of a family of statistics, namely the power sum symmetric polynomials for the vector of weights of the process, as the concentration parameters tend to infinity. We establish a central limit theorem and obtain explicit representations for the asymptotic variance, with the latter clearly showing the impact of each component in the hierarchical structure. These results are crucial for understanding the asymptotic behavior of the sampling formulas associated with the process. In comparison with the known results for the hierarchical Dirichlet process, the results for the hierarchical Pitman-Yor process are mathematically more challenging and structurally more revealing of power law behavior.

math.PR↗

arXiv2025

Asymptotic behavior of clusters in hierarchical species sampling models

Consider a random sample of size $N$ from a hierarchical species sampling model. In this paper, we study the large $N$ asymptotic behavior of the number ${\bf K}_N$ of clusters in the random sample and the number ${\bf \widetilde M}_{\ell,N}$ of clusters represented by exactly $\ell$ latent first-level clusters in the first level of the hierarchical model. In particular, we establish almost sure and $L^p$ convergence for ${\bf \widetilde M}_{\ell,N}$, Gaussian fluctuations and the law of the iterated logarithm for ${\bf K}_N$, and large deviation principles for both ${\bf K}_N$ and ${\bf \widetilde M}_{\ell,N}$. Our approach relies on a random sample size (or random index) representation of the number of clusters through the corresponding non-hierarchical species sampling model.

math.PR↗

arXiv2024

Central limit theorems associated with the hierarchical Dirichlet process

The hierarchical Dirichlet process is a discrete random measure used as a prior in Bayesian nonparametrics and motivated by the study of groups of clustered data. We study the asymptotic behavior of the power sum symmetric polynomials for the vector of weights of the hierarchical Dirichlet process as the concentration parameters tend to infinity. We establish central limit theorems and obtain explicit representations for the asymptotic variances, with the latter clearly showing the impact of the hierarchical structure. These objects are related to the homozygosity in population genetics, the Simpson diversity index in ecology, and the Herfindahl-Hirschman index in economics.

math.PR↗

arXiv2024

Large parameter asymptotic analysis for homogeneous normalized random measures with independent increments

Homogeneous normalized random measures with independent increments (hNRMIs) represent a broad class of Bayesian nonparametric priors and thus are widely used. In this paper, we obtain the strong law of large numbers, the central limit theorem and the functional central limit theorem of hNRMIs when the concentration parameter $a$ approaches infinity. To quantify the convergence rate of the obtained central limit theorem, we further study the Berry-Esseen bound, which turns out to be of the form $O \left( \frac{1}{\sqrt{a}}\right)$. As an application of the central limit theorem, we present the functional delta method, which can be employed to obtain the limit of the quantile process of hNRMIs. As an illustration of the central limit theorems, we demonstrate the convergence numerically for the Dirichlet processes and the normalized inverse Gaussian processes with various choices of the concentration parameters.

math.ST↗

arXiv2022

Hierarchical Dirichlet Process and Relative Entropy

The Hierarchical Dirichlet process is a discrete random measure serving as an important prior in Bayesian non-parametrics. It is motivated with the study of groups of clustered data. Each group is modelled through a level two Dirichlet process and all groups share the same base distribution which itself is a drawn from a level one Dirichlet process. It has two concentration parameters with one at each level. The main results of the paper are the law of large numbers and large deviations for the hierarchical Dirichlet process and its mass when both concentration parameters converge to infinity. The large deviation rate functions are identified explicitly. The rate function for the hierarchical Dirichlet process consists of two terms corresponding to the relative entropies at each level. It is less than the rate function for the Dirichlet process, which reflects the fact that the number of clusters under the hierarchical Dirichlet process has a slower growth rate than under the Dirichlet process.

math.PR↗

arXiv2022

Strategy-Driven Limit Theorems Associated Bandit Problems

Motivated by the study of asymptotic behaviour of the bandit problems, we obtain several strategy-driven limit theorems including the law of large numbers, the large deviation principle, and the central limit theorem. Different from the classical limit theorems, we develop sampling strategy-driven limit theorems that generate the maximum or minimum average reward. The law of large numbers identifies all possible limits that are achievable under various strategies. The large deviation principle provides the maximum decay probabilities for deviations from the limiting domain. To describe the fluctuations around averages, we obtain strategy-driven central limit theorems under optimal strategies. The limits in these theorem are identified explicitly, and depend heavily on the structure of the events or the integrating functions and strategies. This demonstrates the key signature of the learning structure. Our results can be used to estimate the maximal (minimal) rewards, and to identify the conditions of avoiding the Parrondo's paradox in the two-armed bandit problem. It also lays the theoretical foundation for statistical inference in determining the arm that offers the higher mean reward.

math.PR↗

arXiv2021

Large Deviations For Randomly Weighted Sums of Random Measures

Let $\{{\bf \mathcal{Z}}_n:n\geq 1\}$ be a sequence of i.i.d. random probability measures. Independently, for each $n\geq 1$, let $(X_{n1},\ldots, X_{nn})$ be a random vector of positive random variables that add up to one. This paper studies the large deviation principles for the randomly weighted sum $\sum_{i=1}^{n} X_{ni} \mathcal{Z}_i$. In the case of finite Dirichlet weighted sum of Dirac measures, we obtain an explicit form for the rate function. It provides a new measurement of divergence between probabilities. As applications, we obtain the large deviation principles for a class of randomly weighted means including the Dirichlet mean and the corresponding posterior mean. We also identify the minima of relative entropy with mean constraint in both forward and reverse directions.

math.PR↗

arXiv2018

Bayesian nonparametric analysis of Kingman's coalescent

Kingman's coalescent is one of the most popular models in population genetics. It describes the genealogy of a population whose genetic composition evolves in time according to the Wright-Fisher model, or suitable approximations of it belonging to the broad class of Fleming-Viot processes. Ancestral inference under Kingman's coalescent has had much attention in the literature, both in practical data analysis, and from a theoretical and methodological point of view. Given a sample of individuals taken from the population at time $t>0$, most contributions have aimed at making frequentist or Bayesian parametric inference on quantities related to the genealogy of the sample. In this paper we propose a Bayesian nonparametric predictive approach to ancestral inference. That is, under the prior assumption that the composition of the population evolves in time according to a neutral Fleming-Viot process, and given the information contained in an initial sample of $m$ individuals taken from the population at time $t>0$, we estimate quantities related to the genealogy of an additional unobservable sample of size $m^{\prime}\geq1$. As a by-product of our analysis we introduce a class of Bayesian nonparametric estimators (predictors) which can be thought of as Good-Turing type estimators for ancestral inference. The proposed approach is illustrated through an application to genetic data.

stat.ME↗

arXiv2017

A dynamic model for the two-parameter Dirichlet process

Let $\alpha=1/2$, $\theta>-1/2$, and $\nu_0$ be a probability measure on a type space $S$. In this paper, we investigate the stochastic dynamic model for the two-parameter Dirichlet process $\Pi_{\alpha,\theta,\nu_0}$. If $S=\mathbb{N}$, we show that the bilinear form \begin{eqnarray*} \left\{ \begin{array}{l} {\cal E}(F,G)=\frac{1}{2}\int_{{\cal P}_1(\mathbb{N})}\langle \nabla F(\mu),\nabla G(\mu)\rangle_{\mu} \Pi_{\alpha,\theta,\nu_0}(d\mu),\ \ F,G\in {\cal F},\\ {\cal F}=\{F(\mu)=f(\mu(1),\dots,\mu(d)):f\in C^{\infty}(\mathbb{R}^d), d\ge 1\} \end{array} \right. \end{eqnarray*} is closable on $L^2({\cal P}_1(\mathbb{N});\Pi_{\alpha,\theta,\nu_0})$ and its closure $({\cal E}, D({\cal E}))$ is a quasi-regular Dirichlet form. Hence $({\cal E}, D({\cal E}))$ is associated with a diffusion process in ${\cal P}_1(\mathbb{N})$ which is time-reversible with the stationary distribution $\Pi_{\alpha,\theta,\nu_0}$. If $S$ is a general locally compact, separable metric space, we discuss properties of the model \begin{eqnarray*} \left\{ \begin{array}{l} {\cal E}(F,G)=\frac{1}{2}\int_{{\cal P}_1(S)}\langle \nabla F(\mu),\nabla G(\mu)\rangle_{\mu} \Pi_{\alpha,\theta,\nu_0}(d\mu),\ \ F,G\in {\cal F},\\ {\cal F}=\{F(\mu)=f(\langle \phi_1,\mu\rangle,\dots,\langle \phi_d,\mu\rangle): \phi_i\in B_b(S),1\le i\le d,f\in C^{\infty}(\mathbb{R}^d),d\ge 1\}. \end{array} \right. \end{eqnarray*} In particular, we prove the Mosco convergence of its projection forms.

math.PR↗

arXiv2016

Moderate deviations for Ewens-Pitman exchangeable random partitions

Consider a population of individuals belonging to an infinity number of types, and assume that type proportions follow the two-parameter Poisson-Dirichlet distribution. A sample of size n is selected from the population. The total number of different types and the number of types appearing in the sample with a fixed frequency are important statistics. In this paper we establish the moderate deviation principles for these quantities. The corresponding rate functions are explicitly identified, which help revealing a critical scale and understanding the exact role of the parameters. Conditional, or posterior, counterparts of moderate deviation principles are also established.

math.PR↗

arXiv2016

Limit Theorems Associated With The Pitman-Yor Process

The Pitman-Yor process is a random discrete measure. The random weights or masses follow the two-parameter Poisson-Dirichlet distribution with parameters $0<\alpha<1, \theta>-\alpha$. The parameters $\alpha$ and $\theta$ correspond to the stable and gamma components, respectively. The distribution of atoms is given by a probability $\nu$. In this article we consider the limit theorems for the Pitman-Yor process and the two-parameter Poisson-Dirichlet distribution. These include law of large numbers, fluctuations, and moderate or large deviation principles. The limiting procedures involve either $\alpha$ tends to zero or one. They arise naturally in genetics and physics such as the asymptotic coalescence time for explosive branching process and the approximation to generalized random energy model for disordered system.

math.PR↗

arXiv2014

Harnack Inequality and Applications for Infinite-Dimensional GEM Processes

The dimension-free Harnack inequality and uniform heat kernel upper/lower bounds are derived for a class of infinite-dimensional GEM processes, which was introduced in \cite{FW} to simulate the two-parameter GEM distributions. In particular, the associated Dirichlet form satisfies the super log-Sobolev inequality which strengthens the log-Sobolev inequality derived in \cite{FW}. To prove the main results, explicit Harnack inequality and super Poincar\'e inequality are established for the one-dimensional Wright-Fisher diffusion processes. The main tool of the study is the coupling by change of measures.

math.PR↗

arXiv2014

Large deviation principles for the Ewens-Pitman sampling model

Let $M_{l,n}$ be the number of blocks with frequency $l$ in the exchangeable random partition induced by a sample of size $n$ from the Ewens-Pitman sampling model. We show that, as $n$ tends to infinity, $n^{-1}M_{l,n}$ satisfies a large deviation principle and we characterize the corresponding rate function. A conditional counterpart of this large deviation principle is also presented. Specifically, given an initial sample of size $n$ from the Ewens-Pitman sampling model, we consider an additional sample of size $m$. For any fixed $n$ and as $m$ tends to infinity, we establish a large deviation principle for the conditional number of blocks with frequency $l$ in the enlarged sample, given the initial sample. Interestingly, the conditional and unconditional large deviation principles coincide, namely there is no long lasting impact of the given initial sample. Potential applications of our results are discussed in the context of Bayesian nonparametric inference for discovery probabilities.

math.PR↗

arXiv2011

Gamma-Dirichlet Structure and Two Classes of Measure-valued Processes

The Gamma-Dirichlet structure corresponds to the decomposition of the gamma process into the independent product of a gamma random variable and a Dirichlet process. This structure allows us to study the properties of the Dirichlet process through the gamma process and vice versa. In this article, we begin with a brief review of existing results concerning the Gamma-Dirichlet structure. New results are obtained for the large deviations of the jump sizes of the gamma process and the quasi-invariance of the two-parameter Poisson-Dirichlet distribution. The laws of the gamma process and the Dirichlet process are the respective reversible measures of the measure-valued branching diffusion with immigration and the Fleming-Viot process with parent independent mutation. We view the relation between these two classes of measure-valued processes as the dynamical Gamma-Dirichlet structure. Other results of this article include the derivation of the transition function of the Fleming-Viot process with parent independent mutation from the transition function of the measure-valued branching diffusion with immigration, and the establishment of the reversibility of the latter. One of these is related to an open problem by Ethier and Griffiths and the other leads to an alternative proof of the reversibility of the Fleming-Viot process.

math.PR↗

arXiv2009

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 Gamma component respectively. The moderate deviation principles are established for the two-parameter Poisson-Dirichlet distribution and the corresponding homozygosity when $θ$ approaches infinity, and the large deviation principle is established for the two-parameter Poisson-Dirichlet distribution when both $α$ and $θ$ approach zero.

math.PR↗

arXiv2009

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 $Π_{α,θ,ν_0}$ 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 $Π_{α,θ,ν_0}$. The methods used come from the theory of Dirichlet forms.

math.PR↗

arXiv2008

Moderate deviations for Poisson--Dirichlet distribution

The Poisson--Dirichlet distribution arises in many different areas. The parameter $θ$ in the distribution is the scaled mutation rate of a population in the context of population genetics. The limiting case of $θ$ 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 $θ$ 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 $θ$, 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.

math.PR↗