Metropolis-Hastings Algorithm with Delayed Acceptance and Rejection

Metropolis-Hastings algorithms are slowed down by the computation of complex target distributions. To solve this problem, one can use the delayed acceptance Metropolis-Hastings algorithm (MHDA) of Christen and Fox (2005). However, the acceptance rate of a proposed value will always be less than in the standard Metropolis-Hastings. We can fix this problem by using the Metropolis-Hastings algorithm with delayed rejection (MHDR) proposed by Tierney and Mira (1999). In this paper, we combine the ideas of MHDA and MHDR to propose a new MH algorithm, named the Metropolis-Hastings algorithm with delayed acceptance and rejection (MHDAR). The new algorithm reduces the computational cost by division of the prior or likelihood functions and increase the acceptance probability by delay rejection of the second stage. We illustrate those accelerating features by a realistic example.

A Bayesian Mixture Model Approach to Disparity Testing

The topic of this article is one-sided hypothesis testing for disparity, i.e., the mean of one group is larger than that of another when there is uncertainty as to which group a datum is drawn. For each datum, the uncertainty is captured with a given discrete probability distribution over the groups. Such situations arise, for example, in the use of Bayesian imputation methods to assess race and ethnicity disparities with certain insurance, health, and financial data. A widely used method to implement this assessment is the Bayesian Improved Surname Geocoding (BISG) method which assigns a discrete probability over six race/ethnicity groups to an individual given the individual’s surname and address location. Using a Bayesian framework and Markov Chain Monte Carlo sampling from the joint posterior distribution of the group means, the probability of a disparity hypothesis is estimated. Four methods are developed and compared with an illustrative data set. Three of these methods are implemented in an R-code and one method in WinBUGS. These methods are programed for any number of groups between two and six inclusive. All the codes are provided in the appendices.

Robust Frequency Estimation Under Additive Symmetric α-Stable Gaussian Mixture Noise

Here the estimating problem of a single sinusoidal signal in the additive symmetricα-stable Gaussian(ASαSG)noise is investigated.The ASαSG noise here is expressed as the additive of a Gaussian noise and a symmetricα-stable distributed variable.As the probability density function(PDF)of the ASαSG is complicated,traditional estimators cannot provide optimum estimates.Based on the Metropolis-Hastings(M-H)sampling scheme,a robust frequency estimator is proposed for ASαSG noise.Moreover,to accelerate the convergence rate of the developed algorithm,a new criterion of reconstructing the proposal covar-iance is derived,whose main idea is updating the proposal variance using several previous samples drawn in each iteration.The approximation PDF of the ASαSG noise,which is referred to the weighted sum of a Voigt function and a Gaussian PDF,is also employed to reduce the computational complexity.The computer simulations show that the performance of our method is better than the maximum likelihood and the lp-norm estimators.

一种新的贝叶斯调制分类算法

提出了一种基于马尔可夫链蒙特卡罗(MCMC)的数字调制分类方法。针对存在未知残留载波相位和频率时贝叶斯分类难以实现的问题,采用Metropolis-Hastings(M-H)算法估计边缘似然概率密度,从而在分类性能上保持了贝叶斯分类的理论最优性和稳健性。利用对比实验验证了方法的性能。

高斯Copula的多光谱遥感影像分割

为了充分利用多光谱影像波段间的相关性,提出高斯Copula的多光谱遥感影像分割方法.首先,建立基于马尔可夫随机场的标号场模型,使用Potts模型刻画该标号场.然后,建立表征像素光谱测度的特征场,利用高斯Copula建立像素光谱测度的多变量统计模型以刻画该特征场.结合标号场、特征场模型及各模型参数的先验概率,利用贝叶斯定理建立多光谱影像分割的后验概率模型.最后,设计适用于模拟后验概率模型的M-H算法,在最大后验概率策略下获取最优分割结果.对模拟和真实多光谱影像分割结果表明,文中方法描述波段间相关性的能力较强,准确性较高.

Bayesian Study Using MCMC of Three-Parameter Frechet Distribution Based on Type-I Censored Data

Type-I censoring mechanism arises when the number of units experiencing the event is random but the total duration of the study is fixed. There are a number of mathematical approaches developed to handle this type of data. The purpose of the research was to estimate the three parameters of the Frechet distribution via the frequentist Maximum Likelihood and the Bayesian Estimators. In this paper, the maximum likelihood method (MLE) is not available of the three parameters in the closed forms;therefore, it was solved by the numerical methods. Similarly, the Bayesian estimators are implemented using Jeffreys and gamma priors with two loss functions, which are: squared error loss function and Linear Exponential Loss Function (LINEX). The parameters of the Frechet distribution via Bayesian cannot be obtained analytically and therefore Markov Chain Monte Carlo is used, where the full conditional distribution for the three parameters is obtained via Metropolis-Hastings algorithm. Comparisons of the estimators are obtained using Mean Square Errors (MSE) to determine the best estimator of the three parameters of the Frechet distribution. The results show that the Bayesian estimation under Linear Exponential Loss Function based on Type-I censored data is a better estimator for all the parameter estimates when the value of the loss parameter is positive.

Bayesian Inference on Type-Ⅰ Progressively Hybrid Competing Risks Model

In this paper, we construct a Bayesian framework combining Type-Ⅰ progressively hybrid censoring scheme and competing risks which are independently distributed as exponentiated Weibull distribution with one scale parameter and two shape parameters. Since there exist unknown hyper-parameters in prior density functions of shape parameters, we consider the hierarchical priors to obtain the individual marginal posterior density functions,Bayesian estimates and highest posterior density credible intervals. As explicit expressions of estimates cannot be obtained, the componentwise updating algorithm of Metropolis-Hastings method is employed to compute the numerical results. Finally, it is concluded that Bayesian estimates have a good performance.

Comparison of the Bayesian Methods on Interval-Censored Data for Weibull Distribution

This study considers the estimation of Maximum Likelihood Estimator and the Bayesian Estimator of the Weibull distribution with interval-censored data. The Bayesian estimation can’t be used to solve the parameters analytically and therefore Markov Chain Monte Carlo is used, where the full conditional distribution for the scale and shape parameters are obtained via Metropolis-Hastings algorithm. Also Lindley’s approximation is used. The two methods are compared to maximum likelihood counterparts and the comparisons are made with respect to the mean square error (MSE) to determine the best for estimating of the scale and shape parameters.

Robust Frequency Estimation Under Additive Mixture Noise

In many applications such as multiuser radar communications and astrophysical imaging processing,the encountered noise is usually described by the finite sum ofα-stable(1≤α<2)variables.In this paper,a new parameter estimator is developed,in the presence of this new heavy-tailed noise.Since the closed-formPDF of theα-stable variable does not exist exceptα=1 andα=2,we take the sum of the Cauchy(α=1)and Gaussian(α=2)noise as an example,namely,additive Cauchy-Gaussian(ACG)noise.The probability density function(PDF)of the mixed random variable,can be calculated by the convolution of the Cauchy’s PDF and Gaussian’s PDF.Because of the complicated integral in the PDF expression of the ACG noise,traditional estimators,e.g.,maximum likelihood,are analytically not tractable.To obtain the optimal estimates,a new robust frequency estimator is devised by employing the Metropolis-Hastings(M-H)algorithm.Meanwhile,to guarantee the fast convergence of the M-H chain,a new proposal covariance criterion is also devised,where the batch of previous samples are utilized to iteratively update the proposal covariance in each sampling process.Computer simulations are carried out to indicate the superiority of the developed scheme,when compared with several conventional estimators and the Cramér-Rao lower bound.

Comparison of the Sampling Efficiency in Spatial Autoregressive Model

A random walk Metropolis-Hastings algorithm has been widely used in sampling the parameter of spatial interaction in spatial autoregressive model from a Bayesian point of view. In addition, as an alternative approach, the griddy Gibbs sampler is proposed by [1] and utilized by [2]. This paper proposes an acceptance-rejection Metropolis-Hastings algorithm as a third approach, and compares these three algorithms through Monte Carlo experiments. The experimental results show that the griddy Gibbs sampler is the most efficient algorithm among the algorithms whether the number of observations is small or not in terms of the computation time and the inefficiency factors. Moreover, it seems to work well when the size of grid is 100.

On Analysis of the Behrens-Fisher Problem Based on Bayesian Evidence

In this paper we have demonstrated the ability of the new Bayesian measure of evidence of Yin (2012, Computational Statistics, 27: 237-249) to solve both the Behrens-Fisher problem and Lindley's paradox. We have provided a general proof that for any prior which yields a linear combination of two independent t random variables as posterior distribution of the di erence of means, the new Bayesian measure of evidence given that prior will solve Lindleys' paradox thereby serving as a general proof for the works of Yin and Li (2014, Journal of Applied Mathematics, 2014(978691)) and Goltong?and Doguwa (2018, Open Journal of Statistics, 8: 902-914).?Using the Pareto prior as an example, we have shown by the use of?simulation results that the new Bayesian measure of evidence solves?Lindley's paradox.

Bayesian Analysis of the Behrens-Fisher Problem under a Gamma Prior

Yin [1] has developed a new Bayesian measure of evidence for testing a point null hypothesis which agrees with the frequentist p-value thereby, solving Lindley’s paradox. Yin and Li [2] extended the methodology of Yin [1] to the case of the Behrens-Fisher problem by assigning Jeffreys’ independent prior to the nuisance parameters. In this paper, we were able to show both analytically and through the results from simulation studies that the methodology of Yin?[1] solves simultaneously, the Behrens-Fisher problem and Lindley’s paradox when a Gamma prior is assigned to the nuisance parameters.

基于联合聚焦/超分辨模型和M-H算法的雷达图像重建

针对雷达目标图像,基于散射/成像模型,利用Metropolis-Hastings(M-H)迭代算法,给出了一种数字M-H贝叶斯联合聚焦/超分辨重建方法,通过产生一系列描述目标散射截面(RCS)和散焦参数概率分布特征的样本,从而获得目标RCS元和散焦参数的最佳估计,最终实现低分辨率图像的超分辨率重建。以合成与实测图像数据为例,对本文方法进行了演示,同时基于信噪比(SNR)指标,对其重建性能进行了比较和评估。实验结果表明,本文提出的方法对雷达图像重建效果良好,可用于合成孔径雷达、逆合成孔径雷达及实波束成像等雷达图像的重建。

Most Likely Optimal Subsampled Markov Chain Monte Carlo

Markov Chain Monte Carlo(MCMC) requires to evaluate the full data likelihood at different parameter values iteratively and is often computationally infeasible for large data sets. This paper proposes to approximate the log-likelihood with subsamples taken according to nonuniform subsampling probabilities, and derives the most likely optimal(MLO) subsampling probabilities for better approximation. Compared with existing subsampled MCMC algorithm with equal subsampling probabilities,the MLO subsampled MCMC has a higher estimation efficiency with the same subsampling ratio. The authors also derive a formula using the asymptotic distribution of the subsampled log-likelihood to determine the required subsample size in each MCMC iteration for a given level of precision. This formula is used to develop an adaptive version of the MLO subsampled MCMC algorithm. Numerical experiments demonstrate that the proposed method outperforms the uniform subsampled MCMC.

Bayesian Nonlinear Quantile Regression Approach for Longitudinal Ordinal Data

Longitudinal data with ordinal outcomes commonly arise in clinical and social studies,where the purpose of interest is usually quantile curves rather than a simple reference range.In this paper we consider Bayesian nonlinear quantile regression for longitudinal ordinal data through a latent variable.An efficient Metropolis–Hastings within Gibbs algorithm was developed for model fitting.Simulation studies and a real data example are conducted to assess the performance of the proposed method.Results show that the proposed approach performs well.