Convergence rates of the blocked Gibbs sampler with random scan in the Wasserstein metric

ABSTRACT

This paper establishes explicit estimates of convergence rates for the blocked Gibbs sampler with random scan under the Dobrushin conditions. The estimates of convergence in the Wasserstein metric are obtained by taking purely analytic approaches.     

Geometric ergodicity of the Gibbs sampler for the Poisson change-point model

Poisson change-point models have been widely used for modelling inhomogeneous time-series of count data. There are a number of methods available for estimating the parameters in these models using iterative techniques such as MCMC. Many of these techniques share the common problem that there does not seem to be a definitive way of knowing the number of iterations required to obtain sufficient convergence. In this paper, we show that the Gibbs sampler of the Poisson change-point model is geometrically ergodic. Establishing geometric ergodicity is crucial from a practical point of view as it implies the existence of a Markov chain central limit theorem, which can be used to obtain standard error estimates. We prove that the transition kernel is a trace-class operator, which implies geometric ergodicity of the sampler. We then provide a useful application of the sampler to a model for the quarterly driver fatality counts for the state of Victoria, Australia.     

Antithetic Methods for Gibbs Samplers

In this article we propose a modification to the output from Metropolis-within-Gibbs samplers that can lead to substantial reductions in the variance over standard estimates. The idea is simple: at each time step of the algorithm, introduce an extra sample into the estimate that is negatively correlated with the current sample, the rationale being that this provides a two-sample numerical approximation to a Rao–Blackwellized estimate. As the conditional sampling distribution at each step has already been constructed, the generation of the antithetic sample often requires negligible computational effort. Our method is implementable whenever one subvector of the state can be sampled from its full conditional and the corresponding distribution function may be inverted, or the full conditional has a symmetric density. We demonstrate our approach in the context of logistic regression and hierarchical Poisson models. The data and computer code used in this article are available online.     

A theory for the multiset sampler

The multiset sampler (MSS) can be viewed as a new data augmentation scheme and it has been applied successfully to a wide range of statistical inference problems. The key idea of the MSS is to augment the system with a multiset of the missing components, and construct an appropriate joint distribution of the parameters of interest and the missing components to facilitate the inference based on Markov chain Monte Carlo. The standard data augmentation strategy corresponds to the MSS with multiset size one. This paper provides a theoretical comparison of the MSS with different multiset sizes. We show that the MSS converges to the target distribution faster as the multiset size increases. This explains the improvement in convergence rate for the MSS with large multiset sizes over the standard data augmentation scheme.     

On reparametrization and the Gibbs sampler

Gibbs samplers derived under different parametrizations of the target density can have radically different rates of convergence. In this article, we specify conditions under which reparametrization leaves the convergence rate of a Gibbs chain unchanged. An example illustrates how these results can be exploited in convergence rate analyses.     

Analysis of convergence rates of some Gibbs samplers on continuous state spaces

We use a non-Markovian coupling and small modifications of techniques from the theory of finite Markov chains to analyze some Markov chains on continuous state spaces. The first is a generalization of a sampler introduced by Randall and Winkler, and the second a Gibbs sampler on narrow contingency tables.     

Conditioning in Markov Chain Monte Carlo

Abstract

The so-called “Rao-Blackwellized” estimators proposed by Gelfand and Smith do not always reduce variance in Markov chain Monte Carlo when the dependence in the Markov chain is taken into account. An illustrative example is given, and a theorem characterizing the necessary and sufficient condition for such an estimator to always reduce variance is proved.     

Bayesian Analysis of a Two-State Markov Modulated Poisson Process

Abstract

We postulate observations from a Poisson process whose rate parameter modulates between two values determined by an unobserved Markov chain. The theory switches from continuous to discrete time by considering the intervals between observations as a sequence of dependent random variables. A result from hidden Markov models allows us to sample from the posterior distribution of the model parameters given the observed event times using a Gibbs sampler with only two steps per iteration.     

Reversible algorithm of simulating multivariate densities with multi-hump

To simulate a multivariate density with multi-hump, Markov chain Monte Carlo method encounters the obstacle of escaping from one hump to another, since it usually takes extraordinately long time and then becomes practically impossible to perform. To overcome these difficulties, a reversible scheme to generate a Markov chain, in terms of which the simulated density may be successful in rather general cases of practically avoiding being trapped in local humps, was suggested.     

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Most regression modeling is based on traditional mean regression which results in non-robust estimation results for non-normal errors. Compared to conventional mean regression, composite quantile regression (CQR) may produce more robust parameters estimation. Based on a composite asymmetric Laplace distribution (CALD), we build a Bayesian hierarchical model for the weighted CQR (WCQR). The Gibbs sampler algorithm of Bayesian WCQR is developed to implement posterior inference. Finally, the proposed method are illustrated by some simulation studies and a real data analysis.     

Implementing random scan Gibbs samplers

Summary  The Gibbs sampler, being a popular routine amongst Markov chain Monte Carlo sampling methodologies, has revolutionized the application of Monte Carlo methods in statistical computing practice. The performance of the Gibbs sampler relies heavily on the choice of sweep strategy, that is, the means by which the components or blocks of the random vector X of interest are visited and updated. We develop an automated, adaptive algorithm for implementing the optimal sweep strategy as the Gibbs sampler traverses the sample space. The decision rules through which this strategy is chosen are based on convergence properties of the induced chain and precision of statistical inferences drawn from the generated Monte Carlo samples. As part of the development, we analytically derive closed form expressions for the decision criteria of interest and present computationally feasible implementations of the adaptive random scan Gibbs sampler via a Gaussian approximation to the target distribution. We illustrate the results and algorithms presented by using the adaptive random scan Gibbs sampler developed to sample multivariate Gaussian target distributions, and screening test and image data. Research by RL and ZY supported in part by a US National Science Foundation FRG grant 0139948 and a grant from Lawrence Livermore National Laboratory, Livermore, California, USA.     

A Note on Convergence of the Equi-Energy Sampler

This paper studies maximum likelihood estimation for a parameterised elliptic diffusion in a manifold. The focus is on asymptotic properties of maximum likelihood estimates obtained from continuous time observation. These are well known when the underlying manifold is a Euclidean space. However, no systematic study exists in the case of a general manifold. The starting point is to write down the likelihood function and equation. This is achieved using the tools of stochastic differential geometry. Consistency, asymptotic normality and asymptotic optimality of maximum likelihood estimates are then proved, under regularity assumptions. Numerical computation of maximum likelihood estimates is briefly discussed.     

On a proper way to select population failure distribution and a stochastic optimization method in parameter estimation

It is widely accepted that the Weibull distribution plays an important role in reliability applications. The reliability of a product or a system is the probability that the product or the system will still function for a specified time period when operating under some confined conditions. Parameter estimation for the three parameter Weibull distribution has been studied by many researchers in the past. Maximum likelihood has traditionally been the main method of estimation for Weibull parameters along with other recently proposed hybrids of optimization methods. In this paper, we use a stochastic optimization method called the Markov Chain Monte Carlo (MCMC) to carry out the estimation. The method is extremely flexible and inference for any quantity of interest is easily obtained.     

Approximate Predetermined Convergence Properties of the Gibbs Sampler

This article aims to provide a method for approximately predetermining convergence properties of the Gibbs sampler. This is to be done by first finding an approximate rate of convergence for a normal approximation of the target distribution. The rates of convergence for different implementation strategies of the Gibbs sampler are compared to find the best one. In general, the limiting convergence properties of the Gibbs sampler on a sequence of target distributions (approaching a limit) are not the same as the convergence properties of the Gibbs sampler on the limiting target distribution. Theoretical results are given in this article to justify that under conditions, the convergence properties of the Gibbs sampler can be approximated as well. A number of practical examples are given for illustration.     

A markov chain sampler for contingency table exact inference

Summary  In the inference of contingency table, when the cell counts are not large enough for asymptotic approximation, conditioning exact method is used and often computationally impractical for large tables. Instead, various sampling methods can be used. Based on permutation, the Monte Carlo sampling may become again impractical for large tables. For this, existing the Markov chain method is to sample a few elements of the table at each iteration and is inefficient. Here we consider a Markov chain, in which a sub-table of user specified size is updated at each iteration, and it achieves high sampling efficiency. Some theoretical properties of the chain and its applications to some commonly used tables are discussed. As an illustration, this method is applied to the exact test of the Hardy-Weinberg equilibrium in the population genetics context.     

From metropolis to diffusions: Gibbs states and optimal scaling

This paper investigates the behaviour of the random walk Metropolis algorithm in high-dimensional problems. Here we concentrate on the case where the components in the target density is a spatially homogeneous Gibbs distribution with finite range. The performance of the algorithm is strongly linked to the presence or absence of phase transition for the Gibbs distribution; the convergence time being approximately linear in dimension for problems where phase transition is not present. Related to this, there is an optimal way to scale the variance of the proposal distribution in order to maximise the speed of convergence of the algorithm. This turns out to involve scaling the variance of the proposal as the reciprocal of dimension (at least in the phase transition-free case). Moreover, the actual optimal scaling can be characterised in terms of the overall acceptance rate of the algorithm, the maximising value being 0.234, the value as predicted by studies on simpler classes of target density. The results are proved in the framework of a weak convergence result, which shows that the algorithm actually behaves like an infinite-dimensional diffusion process in high dimensions.     

General Methods for Monitoring Convergence of Iterative Simulations

Abstract

We generalize the method proposed by Gelman and Rubin (1992a) for monitoring the convergence of iterative simulations by comparing between and within variances of multiple chains, in order to obtain a family of tests for convergence. We review methods of inference from simulations in order to develop convergence-monitoring summaries that are relevant for the purposes for which the simulations are used. We recommend applying a battery of tests for mixing based on the comparison of inferences from individual sequences and from the mixture of sequences. Finally, we discuss multivariate analogues, for assessing convergence of several parameters simultaneously.     

Monte Carlo approximation through Gibbs output in generalized linear mixed models

Geyer (J. Roy. Statist. Soc. 56 (1994) 291) proposed Monte Carlo method to approximate the whole likelihood function. His method is limited to choosing a proper reference point. We attempt to improve the method by assigning some prior information to the parameters and using the Gibbs output to evaluate the marginal likelihood and its derivatives through a Monte Carlo approximation. Vague priors are assigned to the parameters as well as the random effects within the Bayesian framework to represent a non-informative setting. Then the maximum likelihood estimates are obtained through the Newton Raphson method. Thus, out method serves as a bridge between Bayesian and classical approaches. The method is illustrated by analyzing the famous salamander mating data by generalized linear mixed models.     

SLTBL模型的Bayes分析

利用M arkov cha in M on te C arlo技术对可分离的下三角双线性模型进行B ayes分析.由于参数联合后验密度的复杂性,我们导出了所有的条件后验分布,以便利用G ibbs抽样器方法抽取后验密度的样本.特别地,由于从模型的方向向量的后验分布中直接抽样是困难的,我们特别设计了一个M etropolis-H astings算法以解决该难题.我们用仿真的方法验证了所建议方法的有效性,并成功应用于分析实际数据.