We develop a new Gibbs sampler for a linear mixed model with a Dirichlet process random effect term, which is easily extended to a generalized linear mixed model with a probit link function. Our Gibbs sampler exploits the properties of the multinomial and Dirichlet distributions, and is shown to be an improvement, in terms of operator norm and efficiency, over other commonly used MCMC algorithms. We also investigate methods for the estimation of the precision parameter of the Dirichlet process, finding that maximum likelihood may not be desirable, but a posterior mode is a reasonable approach. Examples are given to show how these models perform on real data. Our results complement both the theoretical basis of the Dirichlet process nonparametric prior and the computational work that has been done to date.
Duke University（杜克大学）；Princeton University（普林斯顿大学）
We develop a generalized method of moments (GMM) approach for fast parameter estimation in a new class of Dirichlet latent variable models with mixed data types. Parameter estimation via GMM has been demonstrated to have computational and statistical advantages over alternative methods, such as expectation maximization, variational inference, and Markov chain Monte Carlo. The key computational advan- tage of our method (MELD) is that parameter estimation does not require instantiation of the latent variables. Moreover, a representational advantage of the GMM approach is that the behavior of th…
This paper develops the use of Dirichlet forms to deliver proofs of optimal scaling results for Markov chain Monte Carlo algorithms (specifically, Metropolis-Hastings random walk samplers) under regularity conditions which are substantially weaker than those required by the original approach (based on the use of infinitesimal generators). The Dirichlet form methods have the added advantage of providing an explicit construction of the underlying infinite-dimensional context. In particular, this enables us directly to establish weak convergence to the relevant infinite-dimensional distributions.
Consider two forms of sampling from a population: (i) drawing $s$ samples of $n$ elements with replacement and (ii) drawing a single sample of $ns$ elements. In this paper, under the setting where the descending order population frequency follows the Poisson--Dirichlet distribution with parameter $θ$, we report that the magnitude relation of the Fisher information, which sample partitions converted from samples (i) and (ii) possess, can change depending on the parameters, $n$, $s$, and $θ$. Roughly speaking, if $θ$ is small relative to $n$ and $s$, the Fisher information of (i) is larger than …
The paper is concerned with constructing pairwise dependence between $m$ random density functions each of which is modeled as a mixture of Dirichlet process model. The key to this is how to create dependencies between random Dirichlet processes. The present paper adopts a plan previously used for creating pairwise dependence, with the simplification that all random Dirichlet processes share the same atoms. Our contention is that for all dependent Dirichlet process models, common atoms are sufficient. We show that by adopting common atoms, it is possible to compute the $L_p$ distances between a…
We introduce a new class of nonparametric prior distributions on the space of continuously varying densities, induced by Dirichlet process mixtures which diffuse in time. These select time-indexed random functions without jumps, whose sections are continuous or discrete distributions depending on the choice of kernel. The construction exploits the widely used stick-breaking representation of the Dirichlet process and induces the time dependence by replacing the stick-breaking components with one-dimensional Wright-Fisher diffusions. These features combine appealing properties of the model, inh…
The class of Dirichlet random vectors is central in numerous probabilistic and statistical applications. The main result of this paper derives the exact tail asymptotics of the aggregated risk of powers of Dirichlet random vectors when the radial component has df in the Gumbel or the Weibull max-domain of attraction. We present further results for the joint asymptotic independence and the max-sum equivalence.
Gamma Tilting Calculus for GGC and Dirichlet means with applications to Linnik processes and Occupation Time Laws for Randomly Skewed Bessel Processes and Bridges
本文发展了GGC和Dirichlet过程平均泛函的一些一般演算。然后，它继续通过调查的积极Linnik随机变量，更一般的随机变量来自一个稳定的从属GGC从属成分，建立各种分布之间的等价关系，这些模型和现象连接到本地时间和占领时间的什么被定义为随机偏斜贝塞尔过程和桥梁。这产生了大量有趣的身份和明确的密度公式，这些模型。随机偏斜贝塞尔过程和桥可以被看作是Barlow，Pitman and Yor（1989）和Pitman and Yor（1997）中的p偏斜过程和桥的随机化，并且可以通过指数倾斜自然产生。作为一个特殊结果，证明了p-斜随机Bessel过程或（广义）桥的占有时间与非平凡随机斜过程的占有时间在分布上是等价的.
Posterior concentration rates for empirical Bayes procedures, with applications to Dirichlet Process mixtures
在本文中，我们提供了一般的条件来检查模型和先验数据依赖先验（或经验贝叶斯方法）的后验集中率。我们的目标是提供与Ghosal和van der Vaart（2007 a）的开创性论文中提供的条件接近的条件。然后，我们将一般定理应用到两个不同的设置：使用Dirichlet过程混合高斯随机变量的密度估计与基础措施取决于一些经验量和估计的强度下的Aalen模型的计数过程。非齐次泊松过程的模拟研究也说明了我们的结果。在前一种情况下，我们也得到一些结果的估计的混合密度和反卷积问题。在后者中，我们提供了一个一般定理后的浓度率计数过程与Aalen乘法强度与先验不依赖于数据。
In the present paper new insights into the study of the Non-central Dirichlet distribution are provided. This latter is the analogue of the Dirichlet distribution obtained by replacing the Chi-Squared random variables involved in its definition by as many non-central ones. Specifically, a novel approach to tackling the analysis of this model is introduced based on a simple conditional density together with a suitable transposition into the non-central framework of a characterizing property of independent Chi-Squared random variables. This approach thus enables to remedy the undeniable mathemat…
Density estimation represents one of the most successful applications of Bayesian nonparametrics. In particular, Dirichlet process mixtures of normals are the gold standard for density estimation and their asymptotic properties have been studied extensively, especially in the univariate case. However a gap between practitioners and the current theoretical literature is present. So far, posterior asymptotic results in the multivariate case are available only for location mixtures of Gaussian kernels with independent prior on the common covariance matrix, while in practice as well as from a conc…
We extend classic characterisations of posterior distributions under Dirichlet process and gamma random measures priors to a dynamic framework. We consider the problem of learning, from indirect observations, two families of time-dependent processes of interest in Bayesian nonparametrics: the first is a dependent Dirichlet process driven by a Fleming-Viot model, and the data are random samples from the process state at discrete times; the second is a collection of dependent gamma random measures driven by a Dawson-Watanabe model, and the data are collected according to a Poisson point process …
Dirichlet kernel density estimation on the simplex with missing data
Imam Mohammad Ibn Saud Islamic University（伊玛目穆罕默德·本·沙特伊斯兰大学）；Université de Tunis El-Manar（突尼斯埃尔曼纳尔大学）；Dalhousie University（达尔豪斯大学）；Université du Québec à Trois-Rivières（魁北克特罗瓦里维尔斯大学）
Discrete random probability measures and the exchangeable random partitions they induce are key tools for addressing a variety of estimation and prediction problems in Bayesian inference. Indeed, many popular nonparametric priors, such as the Dirichlet and the Pitman-Yor process priors, select discrete probability distributions almost surely and, therefore, automatically induce exchangeable random partitions. Here we focus on the family of Gibbs-type priors, a recent and elegant generalization of the Dirichlet and the Pitman-Yor process priors. These random probability measures share propertie…
We establish that Laplace transforms of the posterior Dirichlet process converge to those of the limiting Brownian bridge process in a neighbourhood about zero, uniformly over Glivenko-Cantelli function classes. For real-valued random variables and functions of bounded variation, we strengthen this result to hold for all real numbers. This last result is proved via an explicit strong approximation coupling inequality.
