Markov Chain Sampling Methods for Dirichlet Process Mixture Models

Abstract

This article reviews Markov chain methods for sampling from the posterior distribution of a Dirichlet process mixture model and presents two new classes of methods. One new approach is to make Metropolis—Hastings updates of the indicators specifying which mixture component is associated with each observation, perhaps supplemented with a partial form of Gibbs sampling. The other new approach extends Gibbs sampling for these indicators by using a set of auxiliary parameters. These methods are simple to implement and are more efficient than previous ways of handling general Dirichlet process mixture models with non-conjugate priors.     

Parameter Estimation in Hidden Markov Models With Intractable Likelihoods Using Sequential Monte Carlo

We propose sequential Monte Carlo-based algorithms for maximum likelihood estimation of the static parameters in hidden Markov models with an intractable likelihood using ideas from approximate Bayesian computation. The static parameter estimation algorithms are gradient-based and cover both offline and online estimation. We demonstrate their performance by estimating the parameters of three intractable models, namely the α-stable distribution, g-and-k distribution, and the stochastic volatility model with α-stable returns, using both real and synthetic data.     

Variational Approximation for Mixtures of Linear Mixed Models

Mixtures of linear mixed models (MLMMs) are useful for clustering grouped data and can be estimated by likelihood maximization through the Expectation–Maximization algorithm. A suitable number of components is then determined conventionally by comparing different mixture models using penalized log-likelihood criteria such as Bayesian information criterion. We propose fitting MLMMs with variational methods, which can perform parameter estimation and model selection simultaneously. We describe a variational approximation for MLMMs where the variational lower bound is in closed form, allowing for fast evaluation and develop a novel variational greedy algorithm for model selection and learning of the mixture components. This approach handles algorithm initialization and returns a plausible number of mixture components automatically. In cases of weak identifiability of certain model parameters, we use hierarchical centering to reparameterize the model and show empirically that there is a gain in efficiency in variational algorithms similar to that in Markov chain Monte Carlo (MCMC) algorithms. Related to this, we prove that the approximate rate of convergence of variational algorithms by Gaussian approximation is equal to that of the corresponding Gibbs sampler, which suggests that reparameterizations can lead to improved convergence in variational algorithms just as in MCMC algorithms. Supplementary materials for the article are available online.     

Estimating Mixture of Dirichlet Process Models

Abstract

Current Gibbs sampling schemes in mixture of Dirichlet process (MDP) models are restricted to using “conjugate” base measures that allow analytic evaluation of the transition probabilities when resampling configurations, or alternatively need to rely on approximate numeric evaluations of some transition probabilities. Implementation of Gibbs sampling in more general MDP models is an open and important problem because most applications call for the use of nonconjugate base measures. In this article we propose a conceptual framework for computational strategies. This framework provides a perspective on current methods, facilitates comparisons between them, and leads to several new methods that expand the scope of MDP models to nonconjugate situations. We discuss one in detail. The basic strategy is based on expanding the parameter vector, and is applicable for MDP models with arbitrary base measure and likelihood. Strategies are also presented for the important class of normal-normal MDP models and for problems with fixed or few hyperparameters. The proposed algorithms are easily implemented and illustrated with an application.     

基于Dirichlet分布有限混合模型的Bayes聚类

本文基于Dirichlet分布有限混合模型,提出了一种用于成分数据的Bayes聚类方法.采用EM算法获得模型参数的估计,用BIC准则确定类数,用类似于Bayes判别的方法对各观测分类.推导了计算公式,编写出程序.模拟研究结果表明,本文提出的方法有较好的聚类效果.     

A Computational Approach for Full Nonparametric Bayesian Inference Under Dirichlet Process Mixture Models

Widely used parametric generalized linear models are, unfortunately, a somewhat limited class of specifications. Nonparametric aspects are often introduced to enrich this class, resulting in semiparametric models. Focusing on single or k-sample problems, many classical nonparametric approaches are limited to hypothesis testing. Those that allow estimation are limited to certain functionals of the underlying distributions. Moreover, the associated inference often relies upon asymptotics when nonparametric specifications are often most appealing for smaller sample sizes. Bayesian nonparametric approaches avoid asymptotics but have, to date, been limited in the range of inference. Working with Dirichlet process priors, we overcome the limitations of existing simulation-based model fitting approaches which yield inference that is confined to posterior moments of linear functionals of the population distribution. This article provides a computational approach to obtain the entire posterior distribution for more general functionals. We illustrate with three applications: investigation of extreme value distributions associated with a single population, comparison of medians in a k-sample problem, and comparison of survival times from different populations under fairly heavy censoring.     

Semiparametric Modeling and Estimation of Instrumental Variable Models

We apply Bayesian methods to a model involving a binary nonrandom treatment intake variable and an instrumental variable in which the functional forms of some of the covariates in both the treatment intake and outcome distributions are unknown. Continuous and binary response variables are considered. Under the assumption that the functional form is additive in the covariates, we develop efficient Markov chain Monte Carlo-based approaches for summarizing the posterior distribution and for comparing various alternative models via marginal likelihoods and Bayes factors. We show in a simulation experiment that the methods are capable of recovering the unknown functions and are sensitive neither to the sample size nor to the degree of confounding as measured by the correlation between the errors in the treatment and response equations. In the binary response case, however, estimation of the average treatment effect requires larger sample sizes, especially when the degree of confounding is high. The methods are applied to an example dealing with the effect on wages of more than 12 years of education.     

Inference for the Number of Topics in the Latent Dirichlet Allocation Model via Bayesian Mixture Modeling

In latent Dirichlet allocation, the number of topics, T, is a hyperparameter of the model that must be specified before one can fit the model. The need to specify T in advance is restrictive. One way of dealing with this problem is to put a prior on T, but unfortunately the distribution on the latent variables of the model is then a mixture of distributions on spaces of different dimensions, and estimating this mixture distribution by Markov chain Monte Carlo is very difficult. We present a variant of the Metropolis–Hastings algorithm that can be used to estimate this mixture distribution, and in particular the posterior distribution of the number of topics. We evaluate our methodology on synthetic data and compare it with procedures that are currently used in the machine learning literature. We also give an illustration on two collections of articles from Wikipedia. Supplemental materials for this article are available online.     

Bayesian Nonparametric Analysis for a Generalized Dirichlet Process Prior

This paper considers a generalization of the Dirichlet process which is obtained by suitably normalizing superposed independent gamma processes having increasing integer-valued scale parameter. A comprehensive treatment of this random probability measure is provided. We prove results concerning its finite-dimensional distributions, moments, predictive distributions and the distribution of its mean. Most expressions are given in terms of multiple hypergeometric functions, thus highlighting the interplay between Bayesian Nonparametrics and special functions. Finally, a suitable simulation algorithm is applied in order to compute quantities of statistical interest.     

A Visualization Technique for Studying the Iterative Estimation of Mixture Densities

Abstract

This article focuses on recent work that analyzes the expectation maximization (EM) evolution of mixtures-based estimators. The goal of this research is the development of effective visualization techniques to portray the mixture model parameters as they change in time. This is an inherently high-dimensional process. Techniques are presented that portray the time evolution of univariate, bivariate, and trivariate finite and adaptive mixtures estimators. Adaptive mixtures is a recently developed variable bandwidth kernel estimator where each of the kernels is not constrained to reside at a sample location. The future role of these techniques in developing new versions of the adaptive mixtures procedure is also discussed.     

Computing Normalizing Constants for Finite Mixture Models via Incremental Mixture Importance Sampling (IMIS)

This article proposes a method for approximating integrated likelihoods in finite mixture models. We formulate the model in terms of the unobserved group memberships, z, and make them the variables of integration. The integral is then evaluated using importance sampling over the z. We propose an adaptive importance sampling function which is itself a mixture, with two types of component distributions, one concentrated and one diffuse. The more concentrated type of component serves the usual purpose of an importance sampling function, sampling mostly group assignments of high posterior probability. The less concentrated type of component allows for the importance sampling function to explore the space in a controlled way to find other, unvisited assignments with high posterior probability. Components are added adaptively, one at a time, to cover areas of high posterior probability not well covered by the current importance sampling function. The method is called incremental mixture importance sampling (IMIS).

IMIS is easy to implement and to monitor for convergence. It scales easily for higher dimensional mixture distributions when a conjugate prior is specified for the mixture parameters. The simulated values on which the estimate is based are independent, which allows for straightforward estimation of standard errors. The self-monitoring aspects of the method make it easier to adjust tuning parameters in the course of estimation than standard Markov chain Monte Carlo algorithms. With only small modifications to the code, one can use the method for a wide variety of mixture distributions of different dimensions. The method performed well in simulations and in mixture problems in astronomy and medical research.     

基于Gibbs抽样算法的贝叶斯动态面板数据模型分析

针对现有动态面板数据分析中存在偶发参数和没有考虑模型参数的不确定性风险问题,提出了基于Gibbs抽样算法的贝叶斯随机系数动态面板数据模型.假设初始值服从平稳分布,自回归系数服从Logit正态分布的条件下,设计了Markov链Monte Carlo数值计算程序,得到了模型参数的贝叶斯估计值.实证研究结果表明:基于Gibb...     

用于指数可靠性增长模型的一类新的先验分布

该文提出了可用于指数分布产品四种可靠性增长试验方案的一类新的先验分布. 这类先验分布以条件分布形式给出, 它适合可靠性增长试验中的各种情况. 各阶段的条件均值和条件方差的表达式被获得, 先验分布的形式与它们的参数间的关系被讨论. 这些结果有助于与专家意见相结合.本文还给出试验末尾产品可靠性的后验密度, Bayesian估计和Bayesian下限.     

基于贝叶斯推断的TAR模型的门限非线性检验

在正态分布的假定下,变点问题按照均值和方差的变化有四种情形.本文把TAR模型门限非线性的检验问题,看作是对应均值变化,方差不变情形下的变点问题.然后利用可逆跳马尔可夫蒙特卡罗模拟(RJMCMC)方法计算两个比较模型(AR和TAR模型)的后验概率.后验概率的结果支持TAR模型表明门限非线性的存在.模拟实验的结果说明基于贝叶斯推断的检验方法可以很好的区分AR和TAR模型.     

Sparse Variational Analysis of Linear Mixed Models for Large Data Sets

It is increasingly common to be faced with longitudinal or multi-level data sets that have large numbers of predictors and/or a large sample size. Current methods of fitting and inference for mixed effects models tend to perform poorly in such settings. When there are many variables, it is appealing to allow uncertainty in subset selection and to obtain a sparse characterization of the data. Bayesian methods are available to address these goals using Markov chain Monte Carlo (MCMC), but MCMC is very computationally expensive and can be infeasible in large p and/or large n problems. As a fast approximate Bayes solution, we recommend a novel approximation to the posterior relying on variational methods. Variational methods are used to approximate the posterior of the parameters in a decomposition of the variance components, with priors chosen to obtain a sparse solution that allows selection of random effects. The method is evaluated through a simulation study, and applied to an epidemiological application.     

Relevance of the Bayesian paradigm for “applied probabilists”

This paper is based on an invited lecture given by the author at the ORSA/TIMS Special Interest Group on Applied Probability Conference onStatistical and Computational Problems in Probability Modeling, held at Williamsburg, Virginia, January 7–9, 1985.The theme of this paper is twofold. First, that members of the above group should be seriously concerned with issues of statistical inference — they should not stop short upon proposing a probability model. Second, that inference be undertaken via a strict adherence to the rules of probability — the Bayesian paradigm. To underscore a need for emphasizing the first theme, it may be pertinent to note that an overwhelming majority of the papers dealing with statistical and inferential issues that were presented at this conference were authored by members who did not claim to belong to the ORSA/TIMS Special Interest Group on Applied Probability.The lecture was followed by a panel discussion, with Drs. Lyle Broemeling and Edward Wegman of the Office of Naval Research as discussants. Dr. Robert Launer of the Army Research Office served as a moderator. Discussions from the floor included comments by Professors D. Harrington of Harvard University, E. Parzen of Texas A & M University, and R. Smith of Imperial College, London, England. This paper, and the comments of the panelists, are published in this volume of theAnnals of Operations Research, which is going to serve as a Proceedings of the Conference.Supported by Contract No. N00014-85-K-0202, Office of Naval Research, and Grant No. DAAG 29-84-K-0160, Army Research Office.     

Pitfalls of Fitting Autoregressive Models for Heavy-Tailed Time Series

We consider the analysis of time series data which require models with a heavy-tailed marginal distribution. A natural model to attempt to fit to time series data is an autoregression of order p, where p itself is often determined from the data. Several methods of parameter estimation for heavy tailed autoregressions have been considered, including Yule–Walker estimation, linear programming estimators, and periodogram based estimators. We investigate the statistical pitfalls of the first two methods when the models are mis-specified—either completely or due to the presence of outliers. We illustrate the results of our considerations on both simulated and real data sets. A warning is sounded against the assumption that autoregressions will be an applicable class of models for fitting heavy tailed data.     

A Component-Wise EM Algorithm for Mixtures

Maximum likelihood estimation in finite mixture distributions is typically approached as an incomplete data problem to allow application of the expectation-maximization (EM) algorithm. In its general formulation, the EM algorithm involves the notion of a complete data space, in which the observed measurements and incomplete data are embedded. An advantage is that many difficult estimation problems are facilitated when viewed in this way. One drawback is that the simultaneous update used by standard EM requires overly informative complete data spaces, which leads to slow convergence in some situations. In the incomplete data context, it has been shown that the use of less informative complete data spaces, or equivalently smaller missing data spaces, can lead to faster convergence without sacrifying simplicity. However, in the mixture case, little progress has been made in speeding up EM. In this article we propose a component-wise EM for mixtures. It uses, at each iteration, the smallest admissible missing data space by intrinsically decoupling the parameter updates. Monotonicity is maintained, although the estimated proportions may not sum to one during the course of the iteration. However, we prove that the mixing proportions will satisfy this constraint upon convergence. Our proof of convergence relies on the interpretation of our procedure as a proximal point algorithm. For performance comparison, we consider standard EM as well as two other algorithms based on missing data space reduction, namely the SAGE and AECME algorithms. We provide adaptations of these general procedures to the mixture case. We also consider the ECME algorithm, which is not a data augmentation scheme but still aims at accelerating EM. Our numerical experiments illustrate the advantages of the component-wise EM algorithm relative to these other methods.     

Adaptive Methods for Spatial Scan Analysis via Semiparametric Mixture Models

Spatial scan density (SSD) estimation via mixture models is an important problem in the field of spatial statistical analysis and has wide applications in image analysis. The “borrowed strength” density estimation (BSDE) method via mixture models enables one to estimate the local probability density function in a random field wherein potential similarities between the density functions for the subregions are exploited. This article proposes an efficient methods for SSD estimation by integrating the borrowed strength technique into the alternative EM framework which combines the statistical basis of the BSDE approach with the stability and improved convergence rate of the alternative EM methods. In addition, we propose adaptive SSD estimation methods that extend the aforementioned approach by eliminating the need to find the posterior probability of membership of the component densities afresh in each subregion. Simulation results and an application to the detection and identification of man-made regions of interest in an unmanned aerial vehicle imagery experiment show that the adaptive methods significantly outperform the BSDE method. Other applications include automatic target recognition, mammographic image analysis, and minefield detection.     

混料模型D-最优试验设计的改进DE算法研究

差分进化算法(Differential Evolution Algorithm)有较强的全局收敛能力和稳健性,应用该算法可解决工程学、计算机科学等领域的一些复杂优化问题。本文介绍一个改进DE算法,该算法适用于混料模型的近似最优设计问题,对于具有附加约束试验域的混料问题也可高效求解。最后,本文给出应用改进DE算法求解混料模型D-最优试验设计的例子。