检索结果-内蒙古大学图书馆

A Hybrid Scan Gibbs Sampler for Bayesian Models with Latent Variables

STATISTICAL SCIENCE 2021年第3期36卷 379-399页

作者： Backlund, Grant Hobert, James P. Jung, Yeun Ji Khare, Kshitij Univ Florida Dept Stat Gainesville FL 32611 USA JP Morgan Chase & Co Model Governance Grp New York NY 10172 USA

Gibbs sampling is a widely popular Markov chain Monte Carlo algorithm that can be used to analyze intractable posterior distributions associated with Bayesian hierarchical models. There are two standard versions of the Gibbs sampler: The systematic scan (SS) version, where all variables are updated at each iteration, and the random scan (RS) version, where a single, randomly selected variable is updated at each iteration. The literature comparing the theoretical properties of SS and RS Gibbs samplers is reviewed, and an alternative hybrid scan Gibbs sampler is introduced, which is particularly well suited to Bayesian models with latent variables. The word "hybrid" reflects the fact that the scan used within this algorithm has both systematic and random elements. Indeed, at each iteration, one updates the entire set of latent variables, along with a randomly chosen block of the remaining variables. The hybrid scan (HS) Gibbs sampler has important advantages over the two standard scan Gibbs samplers. First, the HS algorithm is often easier to analyze from a theoretical standpoint. In particular, it can be much easier to establish the geometric ergodicity of a HS Gibbs Markov chain than to do the same for the corresponding SS and RS versions. Second, the sandwich methodology developed in (Ann. Statist. 36 (2008) 532-554), which is also reviewed, can be applied to the HS Gibbs algorithm (but not to the standard scan Gibbs samplers). It is shown that, under weak regularity conditions, adding sandwich steps to the HS Gibbs sampler always results in a theoretically superior algorithm. Three specific Bayesian hierarchical models of varying complexity are used to illustrate the results. One is a simple location-scale model for data from the Student's t distribution, which is used as a pedagogical tool. The other two are sophisticated, yet practical Bayesian regression models.

关键词： Geometric ergodicity heavy-tailed errors linear mixed model Markov chain Monte Carlo sandwich algorithm shrinkage prior

来源：评论

学校读者我要写书评

暂无评论

A numerical method for the approximation of reachable sets of linear control systems

引用

IMA JOURNAL OF MATHEMATICAL CONTROL AND INFORMATION 2019年第2期36卷 423-441页

作者： Shao, Lizhen Zhao, Fangyuan Hu, Guangda Univ Sci & Technol Beijing Sch Automat & Elect Engn Beijing 100083 Peoples R China Shanghai Univ Dept Math Shanghai 200444 Peoples R China

In this article, a numerical method for the approximation of reachable sets of linear control systems is discussed. First a continuous system is transformed into a discrete one with Runge-Kutta methods. Then based on Benson's outer approximation algorithm for solving multiobjective optimization problems, we propose a variant of Benson's algorithm to sandwich the reachable set of the discrete system with an inner approximation and an outer approximation. By specifying an approximation error, the quality of the approximations measured in Hausdorff distance can be directly controlled. Furthermore, we use an illustrative example to demonstrate the working of the algorithm. Finally, computational experiments illustrate the superior performance of our proposed algorithm compared to a recent algorithm in the literature.

关键词： reachable set linear system convex programming sandwich algorithm

来源：评论

学校读者我要写书评

暂无评论

Trace Class Markov Chains for Bayesian Inference with Generalized Double Pareto Shrinkage Priors

引用

SCANDINAVIAN JOURNAL OF STATISTICS 2017年第2期44卷 307-323页

作者： Pal, Subahdip Khare, Kshitij Hobert, James P. Emory Univ Dept Biostat Atlanta GA 30322 USA Univ Florida Dept Stat Gainesville FL 32611 USA

Bayesian shrinkage methods have generated a lot of interest in recent years, especially in the context of high-dimensional linear regression. In recent work, a Bayesian shrinkage approach using generalized double Pareto priors has been proposed. Several useful properties of this approach, including the derivation of a tractable three-block Gibbs sampler to sample from the resulting posterior density, have been established. We show that the Markov operator corresponding to this three-block Gibbs sampler is not Hilbert-Schmidt. We propose a simpler two-block Gibbs sampler and show that the corresponding Markov operator is trace class (and hence Hilbert-Schmidt). Establishing the trace class property for the proposed two-block Gibbs sampler has several useful consequences. Firstly, it implies that the corresponding Markov chain is geometrically ergodic, thereby implying the existence of a Markov chain central limit theorem, which in turn enables computation of asymptotic standard errors for Markov chain-based estimates of posterior quantities. Secondly, because the proposed Gibbs sampler uses two blocks, standard recipes in the literature can be used to construct a sandwich Markov chain (by inserting an appropriate extra step) to gain further efficiency and to achieve faster convergence. The trace class property for the two-block sampler implies that the corresponding sandwich Markov chain is also trace class and thereby geometrically ergodic. Finally, it also guarantees that all eigenvalues of the sandwich chain are dominated by the corresponding eigenvalues of the Gibbs sampling chain (with at least one strict domination). Our results demonstrate that a minor change in the structure of a Markov chain can lead to fundamental changes in its theoretical properties. We illustrate the improvement in efficiency resulting from our proposed Markov chains using simulated and real examples.

关键词： Bayesian shrinkage double Pareto prior geometric ergodicity sandwich algorithm trace class operator

来源：评论

学校读者我要写书评

暂无评论

A comparison theorem for data augmentation algorithms with applications

引用

ELECTRONIC JOURNAL OF STATISTICS 2016年第1期10卷 308-329页

作者： Choi, Hee Min Hobert, James P. Univ Calif Davis Dept Stat Davis CA 95616 USA Univ Florida Dept Stat Gainesville FL 32611 USA

The data augmentation (DA) algorithm is considered a useful Markov chain Monte Carlo algorithm that sometimes suffers from slow convergence. It is often possible to convert a DA algorithm into a sandwich algorithm that is computationally equivalent to the DA algorithm, but converges much faster. Theoretically, the reversible Markov chain that drives the sandwich algorithm is at least as good as the corresponding DA chain in terms of performance in the central limit theorem and in the operator norm sense. In this paper, we use the sandwich machinery to compare two DA algorithms. In particular, we provide conditions under which one DA chain can be represented as a sandwich version of the other. Our results are used to extend Hobert and Marchev's (2008) results on the Haar PX-DA algorithm and to improve the collapsing theorem of Liu et al. (1994) and Liu (1994). We also illustrate our results using Brownlee's (1965) stack loss data.

关键词： Data augmentation algorithm sandwich algorithm central limit theorem convergence rate operator norm

来源：评论

学校读者我要写书评

暂无评论

Improving the Data Augmentation algorithm in the Two-Block Setup

引用

JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 2015年第4期24卷 1114-1133页

作者： Pal, Subhadip Khare, Kshitij Hobert, James P. Univ Florida Dept Stat Gainesville FL 32611 USA

The data augmentation (DA) approach to approximate sampling from an intractable probability density f(X) is based on the construction of a joint density, f(X, Y), whose conditional densities, f(X|Y) and f(Y|X), can be straightforwardly sampled. However, many applications of the DA algorithm do not fall in this single-block setup. In these applications, X is partitioned into two components, X = (U, V), in such a way that it is easy to sample from f(Y|X), f(U|V, Y), and f(V|U, Y). We refer to this alternative version of DA, which is effectively a three-variable Gibbs sampler, as two-block DA. We develop two methods to improve the performance of the DA algorithm in the two-block setup. These methods are motivated by the Haar PX-DA algorithm, which has been developed in previous literature to improve the performance of the single-block DA algorithm. The Haar PX-DA algorithm, which adds a computationally inexpensive extra step in each iteration of the DA algorithm while preserving the stationary density, has been shown to be optimal among similar techniques. However, as we illustrate, the Haar PX-DA algorithm does not lead to the required stationary density f(X) in the two-block setup. Our methods incorporate suitable generalizations and modifications to this approach, and work in the two-block setup. A theoretical comparison of our methods to the two-block DA algorithm, a much harder task than the single-block setup due to nonreversibility and structural complexities, is provided. We successfully apply our methods to applications of the two-block DA algorithm in Bayesian robit regression and Bayesian quantile regression. Supplementary materials for this article are available online.

关键词： Data augmentation algorithm Group action Haar measure sandwich algorithm Two-block DA algorithm

来源：评论

学校读者我要写书评

暂无评论

Spectral properties of MCMC algorithms for Bayesian linear regression with generalized hyperbolic errors

引用

STATISTICS & PROBABILITY LETTERS 2014年第0期95卷 92-100页

作者： Jung, Yeun Ji Hobert, James P. Univ Florida Dept Stat Gainesville FL 32611 USA

We study MCMC algorithms for Bayesian analysis of a linear regression model with generalized hyperbolic errors. The Markov operators associated with the standard data augmentation algorithm and a sandwich variant of t... 详细信息

关键词： Bessel function Data augmentation algorithm Geometric convergence rate Markov operator sandwich algorithm Trace-class operator

来源：评论

学校读者我要写书评

暂无评论

Analysis of MCMC algorithms for Bayesian linear regression with Laplace errors

引用

JOURNAL OF MULTIVARIATE ANALYSIS 2013年 117卷 32-40页

作者： Choi, Hee Min Hobert, James P. Univ Florida Dept Stat Gainesville FL 32611 USA

Let pi denote the intractable posterior density that results when the standard default prior is placed on the parameters in a linear regression model with iid Laplace errors. We analyze the Markov chains underlying two different Markov chain Monte Carlo algorithms for exploring pi. In particular, it is shown that the Markov operators associated with the data augmentation (DA) algorithm and a sandwich variant are both trace-class. Consequently, both Markov chains are geometrically ergodic. It is also established that for each i is an element of (1, 2, 3, ...}, the ith largest eigenvalue of the sandwich operator is less than or equal to the corresponding eigenvalue of the DA operator. It follows that the sandwich algorithm converges at least as fast as the DA algorithm. (C) 2013 Elsevier Inc. All rights reserved.

关键词： Asymmetric Laplace distribution Data augmentation algorithm Eigenvalues Geometric convergence rate Markov chain Markov operator Monte Carlo sandwich algorithm Trace-class operator

来源：评论

学校读者我要写书评

暂无评论

A Method for Approximating Univariate Convex Functions Using Only Function Value Evaluations

引用

INFORMS JOURNAL ON COMPUTING 2011年第4期23卷 591-604页

作者： Siem, A. Y. D. den Hertog, D. Hoffmann, A. L. Tilburg Univ Dept Econometr & Operat Res Ctr Econ Res CentER NL-5000 LE Tilburg Netherlands Radboud Univ Nijmegen Dept Radiat Oncol Med Ctr NL-6525 GA Nijmegen Netherlands

In this paper, piecewise-linear upper and lower bounds for univariate convex functions are derived that are only based on function value information. These upper and lower bounds can be used to approximate univariate convex functions. Furthermore, new sandwich algorithms are proposed that iteratively add new input data points in a systematic way until a desired accuracy of the approximation is obtained. We show that our new algorithms that use only function value evaluations converge quadratically under certain conditions on the derivatives. Under other conditions, linear convergence can be shown. Some numerical examples that illustrate the usefulness of the algorithm, including a strategic investment model, are given.

关键词： approximation convexity metamodel analysis of algorithms sandwich algorithm

来源：评论

学校读者我要写书评

暂无评论

Improving the Convergence Properties of the Data Augmentation algorithm with an Application to Bayesian Mixture Modeling

引用

STATISTICAL SCIENCE 2011年第3期26卷 332-351页

作者： Hobert, James P. Roy, Vivekananda Robert, Christian P. Univ Florida Dept Stat Gainesville FL 32611 USA Iowa State Univ Dept Stat Ames IA 50011 USA Univ Paris 09 CEREMADE F-75775 Paris 16 France CREST F-92245 Malakoff France

The reversible Markov chains that drive the data augmentation (DA) and sandwich algorithms define self-adjoint operators whose spectra encode the convergence properties of the algorithms. When the target distribution has uncountable support, as is nearly always the case in practice, it is generally quite difficult to get a handle on these spectra. We show that, if the augmentation space is finite, then (under regularity conditions) the operators defined by the DA and sandwich chains are compact, and the spectra are finite subsets of [0, 1). Moreover, we prove that the spectrum of the sandwich operator dominates the spectrum of the DA operator in the sense that the ordered elements of the former are all less than or equal to the corresponding elements of the latter. As a concrete example, we study a widely used DA algorithm for the exploration of posterior densities associated with Bayesian mixture models [J. Roy. Statist. Soc. Ser. B 56 (1994) 363-375]. In particular, we compare this mixture DA algorithm with an alternative algorithm proposed by Fruhwirth-Schnatter [J. Amer. Statist. Assoc. 96 (2001) 194-209] that is based on random label switching.

关键词： Compact operator convergence rate eigenvalue label switching Markov operator Monte Carlo operator norm positive operator reversible Markov chain sandwich algorithm spectrum

来源：评论

学校读者我要写书评

暂无评论

Solving nonconvex planar location problems by finite dominating sets

引用

JOURNAL OF GLOBAL OPTIMIZATION 2000年第2期18卷 195-210页

作者： Carrizosa, E Hamacher, HW Klein, R Nickel, S Univ Sevilla Fac Matemat Seville Spain Univ Kaiserslautern Fachbereich Math D-6750 Kaiserslautern Germany Fern Univ GH Hagen Fachbereich Informat Hagen Germany Inst Techno & Wirtschaftsmath Kaiserslautern Kaiserslautern Germany

It is well-known that some of the classical location problems with polyhedral gauges can be solved in polynomial time by finding a finite dominating set, i.e. a finite set of candidates guaranteed to contain at least one optimal location. In this paper it is first established that this result holds for a much larger class of problems than currently considered in the literature. The model for which this result can be proven includes, for instance, location problems with attraction and repulsion, and location-allocation problems. Next, it is shown that the approximation of general gauges by polyhedral ones in the objective function of our general model can be analyzed with regard to the subsequent error in the optimal objective value. For the approximation problem two different approaches are described, the sandwich procedure and the greedy algorithm. Both of these approaches lead - for fixed epsilon - to polynomial approximation algorithms with accuracy epsilon for solving the general model considered in this paper.

关键词： continuous location polyhedral gauges finite dominating sets approximation sandwich algorithm greedy algorithm

来源：评论

学校读者我要写书评

暂无评论

建议与咨询留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

时间限定

文献类型

馆藏选择

核心期刊

语言

文献类型

帮助

文字说明：

检索规则说明：

检索范例：

分类表

所选分类

限定检索结果

文献类型

馆藏范围

日期分布

学科分类号

主题

机构

作者

语言

请选择保存的检索档案：

请选择收藏分类：

通借通还

建议与咨询 留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

时间限定

文献类型

馆藏选择

核心期刊

语言

文献类型

帮助

文字说明：

检索规则说明：

检索范例：

分类表

所选分类

限定检索结果

文献类型

馆藏范围

日期分布

学科分类号

主题

机构

作者

语言

请选择保存的检索档案： 新增检索档案 确定 取消

请选择收藏分类： 新增自定义分类 确定 取消

通借通还

建议与咨询留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

请选择保存的检索档案：

请选择收藏分类：