Proposal adaptation in simulated annealing for continuous optimization problems

In recent years, adaptive Markov Chain Monte Carlo (MCMC) methods have become a standard tool for Bayesian parameter estimation. In adaptive MCMC, the past iterations are used to tune the proposal distribution of the algorithm. The same adaptation mechanisms can be used in Simulated Annealing (SA), a popular optimization method based on MCMC. The difficulty in using adaptation directly in SA is that the target function changes along the iterations in the annealing process, and the adaptation should keep up with the annealing. In this paper, a mechanism for automatically tuning the proposal distribution in SA is proposed. The approach is based on the Adaptive Metropolis algorithm of Haario et al. (Bernoulli 7(2):223–242, 2001), combined with a weighting mechanism to account for the cooling target. The proposed adaptation mechanism does not add any computational complexity to the problem in terms of objective function evaluations. The effect of adaptation is demonstrated using two benchmark problems, showing that the proposed adaptation mechanism can significantly improve optimization results compared to non-adaptive SA. The approach is presented for continuous optimization problems and generalization to integer and mixed-integer problems is a topic of future research.     

Estimation of parameters in latent class models using fuzzy clustering algorithms

A mixture approach to clustering is an important technique in cluster analysis. A mixture of multivariate multinomial distributions is usually used to analyze categorical data with latent class model. The parameter estimation is an important step for a mixture distribution. Described here are four approaches to estimating the parameters of a mixture of multivariate multinomial distributions. The first approach is an extended maximum likelihood (ML) method. The second approach is based on the well-known expectation maximization (EM) algorithm. The third approach is the classification maximum likelihood (CML) algorithm. In this paper, we propose a new approach using the so-called fuzzy class model and then create the fuzzy classification maximum likelihood (FCML) approach for categorical data. The accuracy, robustness and effectiveness of these four types of algorithms for estimating the parameters of multivariate binomial mixtures are compared using real empirical data and samples drawn from the multivariate binomial mixtures of two classes. The results show that the proposed FCML algorithm presents better accuracy, robustness and effectiveness. Overall, the FCML algorithm has the superiority over the ML, EM and CML algorithms. Thus, we recommend FCML as another good tool for estimating the parameters of mixture multivariate multinomial models.     

Adaptive Markov Chain Monte Carlo for Bayesian Variable Selection

We describe adaptive Markov chain Monte Carlo (MCMC) methods for sampling posterior distributions arising from Bayesian variable selection problems. Point-mass mixture priors are commonly used in Bayesian variable selection problems in regression. However, for generalized linear and nonlinear models where the conditional densities cannot be obtained directly, the resulting mixture posterior may be difficult to sample using standard MCMC methods due to multimodality. We introduce an adaptive MCMC scheme that automatically tunes the parameters of a family of mixture proposal distributions during simulation. The resulting chain adapts to sample efficiently from multimodal target distributions. For variable selection problems point-mass components are included in the mixture, and the associated weights adapt to approximate marginal posterior variable inclusion probabilities, while the remaining components approximate the posterior over nonzero values. The resulting sampler transitions efficiently between models, performing parameter estimation and variable selection simultaneously. Ergodicity and convergence are guaranteed by limiting the adaptation based on recent theoretical results. The algorithm is demonstrated on a logistic regression model, a sparse kernel regression, and a random field model from statistical biophysics; in each case the adaptive algorithm dramatically outperforms traditional MH algorithms. Supplementary materials for this article are available online.     

Bayesian approaches for analyzing earthquake catastrophic risk

Extreme value theory has been widely used in analyzing catastrophic risk. The theory mentioned that the generalized Pareto distribution (GPD) could be used to estimate the limiting distribution of the excess value over a certain threshold; thus the tail behaviors are analyzed. However, the central behavior is important because it may affect the estimation of model parameters in GPD, and the evaluation of catastrophic insurance premiums also depends on the central behavior. This paper proposes four mixture models to model earthquake catastrophic loss and proposes Bayesian approaches to estimate the unknown parameters and the threshold in these mixture models. MCMC methods are used to calculate the Bayesian estimates of model parameters, and deviance information criterion values are obtained for model comparison. The earthquake loss of Yunnan province is analyzed to illustrate the proposed methods. Results show that the estimation of the threshold and the shape and scale of GPD are quite different. Value-at-risk and expected shortfall for the proposed mixture models are calculated under different confidence levels.     

A Gaussian Mixed Model for Learning Discrete Bayesian Networks

In this paper, we address the problem of learning discrete Bayesian networks from noisy data. A graphical model based on a mixture of Gaussian distributions with categorical mixing structure coming from a discrete Bayesian network is considered. The network learning is formulated as a maximum likelihood estimation problem and performed by employing an EM algorithm. The proposed approach is relevant to a variety of statistical problems for which Bayesian network models are suitable—from simple regression analysis to learning gene/protein regulatory networks from microarray data.     

Heteroscedastic normal-exponential mixture models: Bayesian and classical approaches

In this paper, we introduce a Bayesian analysis for mixture of distributions belonging to the exponential family. As a special case we consider a mixture of normal exponential distributions including joint modeling of the mean and variance. We also consider joint modeling of the mean and variance heterogeneity. Markov Chain Monte Carlo (MCMC) methods are used to obtain the posterior summaries of interest. We also introduce and apply an EM algorithm, where the maximization is obtained applying the Fisher scoring algorithm. Finally, we also include analysis of real data sets to illustrate the proposed methodology.     

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.     

Using mixture of Gamma distributions for Bayesian analysis in an M/G/1 queue with optional second service

The paper proposes Bayesian framework in an M/G/1 queuing system with optional second service. The semi-parametric model based on a finite mixture of Gamma distributions is considered to approximate both the general service and re-service times densities in this queuing system. A Bayesian procedure based on birth-death MCMC methodology is proposed to estimate system parameters, predictive densities and some performance measures related to this queuing system such as stationary system size and waiting time. The approach is illustrated with several numerical examples based on various simulation studies.     

Variational Bayesian Generative Topographic Mapping

General finite mixture models are powerful tools for the density-based grouping of multivariate i.i.d. data, but they lack data visualization capabilities, which reduces their practical applicability to real-world problems. Generative topographic mapping (GTM) was originally formulated as a constrained mixture of distributions in order to provide simultaneous visualization and clustering of multivariate data. In its inception, the adaptive parameters were determined by maximum likelihood (ML), using the expectation-maximization (EM) algorithm. The original GTM is, therefore, prone to data overfitting unless a regularization mechanism is included. In this paper, we define an alternative variational formulation of GTM that provides a full Bayesian treatment to a Gaussian process (GP)-based variation of the model. The generalization capabilities of the proposed Variational Bayesian GTM are assessed in some detail and compared with those of alternative GTM regularization approaches in terms of test log-likelihood, using several artificial and real datasets.     

Bayesian Nonparametric Inference for a Multivariate Copula Function

The paper presents a general Bayesian nonparametric approach for estimating a high dimensional copula. We first introduce the skew–normal copula, which we then extend to an infinite mixture model. The skew–normal copula fixes some limitations in the Gaussian copula. An MCMC algorithm is developed to draw samples from the correct posterior distribution and the model is investigated using both simulated and real applications.     

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??Kundu and Gupta proposed to use the importance sampling method to compute the Bayesian estimation of the unknown parameters of the Marshall-Olkin bivariate Weibull distribution. However, we find that the performance of the importance sampling method becomes worse as the sample size gets larger. In this paper, we introduce latent variables to simplify the likelihood function, and use MCMC algorithm to estimate the unknown parameters. Numerical simulations are carried out to assess the performance of the proposed method by comparing with the maximum likelihood estimation, and we find that the Bayesian estimates perform better even for the case of small sample size. A real data is also analyzed for illustrative purpose.     

Implementations of the Monte Carlo EM Algorithm

The Monte Carlo EM (MCEM) algorithm is a modification of the EM algorithm where the expectation in the E-step is computed numerically through Monte Carlo simulations. The most exible and generally applicable approach to obtaining a Monte Carlo sample in each iteration of an MCEM algorithm is through Markov chain Monte Carlo (MCMC) routines such as the Gibbs and Metropolis–Hastings samplers. Although MCMC estimation presents a tractable solution to problems where the E-step is not available in closed form, two issues arise when implementing this MCEM routine: (1) how do we minimize the computational cost in obtaining an MCMC sample? and (2) how do we choose the Monte Carlo sample size? We address the first question through an application of importance sampling whereby samples drawn during previous EM iterations are recycled rather than running an MCMC sampler each MCEM iteration. The second question is addressed through an application of regenerative simulation. We obtain approximate independent and identical samples by subsampling the generated MCMC sample during different renewal periods. Standard central limit theorems may thus be used to gauge Monte Carlo error. In particular, we apply an automated rule for increasing the Monte Carlo sample size when the Monte Carlo error overwhelms the EM estimate at any given iteration. We illustrate our MCEM algorithm through analyses of two datasets fit by generalized linear mixed models. As a part of these applications, we demonstrate the improvement in computational cost and efficiency of our routine over alternative MCEM strategies.     

基于M-H抽样的贝叶斯非对称厚尾GARCH模型研究

针对非对称厚尾GARCH模型参数的预选分布很难确定的问题。对模型参数空间进行数据扩张,把模型中的厚尾残差分布表示成正态分布和逆伽玛分布的混合分布,然后通过对参数的后验条件分布进行变换获得参数的预选分布,从而利用M-H抽样实现了非对称厚尾GARCH模型的贝叶斯分析。中国原油收益率波动的实证研究发现中国原油收益率的波动具有高峰厚尾性但不存在"杠杆效应",样本内的预测评价发现基于M-H抽样的贝叶斯方法优于极大似然方法,说明了M-H抽样方案设计的有效性。     

Expectation-maximization for Bayes-adaptive POMDPs

Bayes-adaptive POMDPs (BAPOMDPs) are partially observable Markov decision problems in which uncertainty in the state-transition and observation-emission probabilities can be captured by a prior distribution over the model parameters. Existing approaches to solving BAPOMDPs rely on model and trajectory sampling to guide exploration and, because of the curse of dimensionality, do not scale well when the degree of model uncertainty is large. In this paper, we begin by presenting two expectation-maximization (EM) approaches to solving BAPOMPs via finite-state controller (FSC) optimization, which at their foundation are extensions of existing EM algorithms for BAMDPs to the more general BAPOMDP setting. The first is a sampling-based EM algorithm that optimizes over a finite number of models drawn from the BAPOMDP prior, and as such is only appropriate for smaller problems with limited model uncertainty; the second approach leverages variational Bayesian methods to ensure tractability without sampling, and is most appropriate for larger domains with greater model uncertainty. Our primary novel contribution is the derivation of the constrained VB-EM algorithm, which addresses an unfavourable preference that often arises towards a certain class of policies when applying the standard VB-EM algorithm. Through an empirical study we show that the sampling-based EM algorithm is competitive with more conventional sampling-based approaches in smaller domains, and that our novel constrained VB-EM algorithm can generate quality solutions in larger domains where sampling-based approaches are no longer viable.     

Model-based clustering and segmentation of time series with changes in regime

Mixture model-based clustering, usually applied to multidimensional data, has become a popular approach in many data analysis problems, both for its good statistical properties and for the simplicity of implementation of the Expectation?CMaximization (EM) algorithm. Within the context of a railway application, this paper introduces a novel mixture model for dealing with time series that are subject to changes in regime. The proposed approach, called ClustSeg, consists in modeling each cluster by a regression model in which the polynomial coefficients vary according to a discrete hidden process. In particular, this approach makes use of logistic functions to model the (smooth or abrupt) transitions between regimes. The model parameters are estimated by the maximum likelihood method solved by an EM algorithm. This approach can also be regarded as a clustering approach which operates by finding groups of time series having common changes in regime. In addition to providing a time series partition, it therefore provides a time series segmentation. The problem of selecting the optimal numbers of clusters and segments is solved by means of the Bayesian Information Criterion. The ClustSeg approach is shown to be efficient using a variety of simulated time series and real-world time series of electrical power consumption from rail switching operations.     

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.     

变量的有限混合模型对有序数据的Bayes聚类分析

本文基于隐变量的有限混合模型, 提出了一种用于有序数据的Bayes聚类方法\bd 我们采用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\bd 模拟研究结果表明, 本文提出的方法有较好的聚类效果, 对于中等规模的数据集, 计算量是可以接受的.     

A Marginal Sampler for σ-Stable Poisson–Kingman Mixture Models

We investigate the class of σ-stable Poisson–Kingman random probability measures (RPMs) in the context of Bayesian nonparametric mixture modeling. This is a large class of discrete RPMs, which encompasses most of the popular discrete RPMs used in Bayesian nonparametrics, such as the Dirichlet process, Pitman–Yor process, the normalized inverse Gaussian process, and the normalized generalized Gamma process. We show how certain sampling properties and marginal characterizations of σ-stable Poisson–Kingman RPMs can be usefully exploited for devising a Markov chain Monte Carlo (MCMC) algorithm for performing posterior inference with a Bayesian nonparametric mixture model. Specifically, we introduce a novel and efficient MCMC sampling scheme in an augmented space that has a small number of auxiliary variables per iteration. We apply our sampling scheme to a density estimation and clustering tasks with unidimensional and multidimensional datasets, and compare it against competing MCMC sampling schemes. Supplementary materials for this article are available online.     

小样本下两阶段MCMC参数估计方法——基于信用风险强度模型的研究

应用我国金融市场数据估计信用风险强度模型参数时,常遇到由小样本而导致的偏差问题,对此本文提出了两阶段MCMC参数估计方法:第一阶段用Lee和Mykland的跳辨识方法估计跳跃项参数;第二阶段用MC-MC方法估计扩散和漂移项参数。误差分析的结果表明两阶段MCMC方法小样本下信用风险模型参数估计的效果要明显好于单纯的MCMC方法。作为应用,采用我国第一支个人住房抵押贷款支持证券"建元2005-1"的违约和提前还款数据,估计了信用风险强度模型的参数。     

Bayesian variable selection in generalized linear models using a combination of stochastic optimization methods

In this paper the usage of a stochastic optimization algorithm as a model search tool is proposed for the Bayesian variable selection problem in generalized linear models. Combining aspects of three well known stochastic optimization algorithms, namely, simulated annealing, genetic algorithm and tabu search, a powerful model search algorithm is produced. After choosing suitable priors, the posterior model probability is used as a criterion function for the algorithm; in cases when it is not analytically tractable Laplace approximation is used. The proposed algorithm is illustrated on normal linear and logistic regression models, for simulated and real-life examples, and it is shown that, with a very low computational cost, it achieves improved performance when compared with popular MCMC algorithms, such as the MCMC model composition, as well as with “vanilla” versions of simulated annealing, genetic algorithm and tabu search.