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The advent of precision cosmology has led to a very accurate observational data. In order to extract cosmological information, we therefore need robust and accurate Cosmic Microwave Background (CMB) reconstruction techniques with reliable statistical error estimates associated with them. We show the performance of our algorithm using simulations of the future Probe of Inflation and Cosmic Origins (PICO) missions. To generate samples from the joint distribution, we employ the internal linear combination (ILC) technique with prior information of CMB E-mode covariance matrix augmented by a Gibbs sampling technique of Sudevan and Saha 2020. We estimate the marginalized densities of $S$ and $C_{\ell }^E$ using the samples from full-posterior. The best-fit cleaned E-mode map and the corresponding angular power spectrum are consistent with the input E-mode map and the sky power spectrum implying accurate reconstruction. Using the samples $C_{\ell }^E$ of all Gibbs chains, we estimate the likelihood function $P(C_{\ell }^E|D$) of any arbitrary $C_{\ell }^E$ given simulated observed maps (data, D) of PICO mission using a Blackwell–Rao estimator. The likelihood function can be seamlessly integrated to the cosmological parameter estimation method. Apart from producing an accurate estimate of E-mode signal over large angular scales, our method also builds a connection between the component reconstruction and reliable cosmological parameter estimation using CMB E-mode observations over large angular scales. The entire method does not assume any explicit models for E-mode foreground components in order to remove them, which is an attractive property since foreground modeling uncertainty does not pose as a challenge in this case.

We provide a tutorial on learning and inference in hidden Markov models in the context of the recent literature on Bayesian networks. This perspective makes it possible to consider novel generalizations of hidden Markov models with multiple hidden state variables, multiscale representations, and mixed discrete and continuous variables. Although exact inference in these generalizations is usually intractable, one can use approximate inference algorithms such as Markov chain sampling and variational methods. We describe how such methods are applied to these generalized hidden Markov models. We conclude this review with a discussion of Bayesian methods for model selection in generalized HMMs.

Nowadays, it is very popular to use deep architectures in machine learning. Deep Belief Networks (DBNs) are deep architectures that use stack of Restricted Boltzmann Machines (RBMs) to create a powerful generative model using the training data. In this paper, we present an improvement in a common method that is usually used in training RBMs. The new method uses free energy as a criterion to obtain elite samples from generative model. We argue that these samples can compute gradient of log probability of training data more accurately. According to the results, an error rate of 0.99% was achieved on MNIST test set. This result shows that the proposed method outperforms the method presented in the first paper introducing DBN (1.25% error rate) and general classification methods such as SVM (1.4% error rate) and KNN (1.6% error rate). In another test using ISOLET dataset, letter classification error dropped to 3.59% compared to 5.59% error rate achieved in the papers using this dataset.

This paper discusses the reliability analysis for a series system in step-stress partially accelerated lifetime test under Type-I progressive hybrid censoring, where independent Burr-XII distributed lifetimes are assumed for the components. In many cases, the exact component causing the system failure cannot be identified and the cause of failure is masked. Bayesian approach combined with auxiliary variables is applied for estimating the parameters of the model when the masking probability is dependent on the component. Further, the reliability and hazard rate functions of the system and components are estimated under use stress level. Simulation studies are performed to demonstrate the efficiency of the methods proposed in this paper under different masking probabilities and different progressive removal schemes.

The three-parameter Burr type XII distribution (3pBXIID) is quite flexible and contains a wide range of distribution shapes for fitting lifetime data. However, it is difficult to obtain reliable estimates of the 3pBXIID quantiles from censored samples for evaluating the reliability of lifetime data. In this work, a Metropolis–Hastings Markov chain Monte Carlo (M-H MCMC) procedure is proposed to obtain reliable maximum likelihood estimates (MLEs) of the 3pBXIID quantiles from a type II censored sample. Moreover, the parametric bootstrap percentile procedure is used to obtain the confidence interval of the quantile of the 3pBXIID. The performance of the proposed M-H MCMC method is evaluated in view of Monte Carlo simulations. Two examples, regarding the survival lifetimes of breast cancer patients and the reliability inference on the lifetimes of oil-well pumps for sucker-rod oil pumping systems, are applied to illustrate the applications of the proposed M-H MCMC method and bootstrap procedure.

The paper considers the Bayesian analysis of an elaborated family of regression models based on the inverse Gaussian distribution, a family that is quite useful for the accelerated test scenario in life testing. A variety of relationships between the model parameters and the entertained stress variable gives rise to four different families. For the Bayesian implementation, the paper considers independent weak priors for the parameters and proposes Gibbs sampler algorithm for obtaining samples from the relevant posteriors. The implementation is also extended when the data are subject to censoring mechanism. Numerical illustration is based on a real dataset on temperature accelerated life testing. Finally, the compatibility of each entertained model is assessed and they are compared using some commonly available Bayesian toolkits. The results are found to be satisfactory.

This paper presents the Bayesian analysis of the stress–strength reliability of a parallel system with active and mixed standby components. The Bayesian estimators of the parameters of exponential and Weibull stress–strength distributions are obtained by assuming informative and noninformative prior for the parameters of the respective distributions. The Gibbs sampling procedure is used to compute the Bayesian stress–strength reliability. The asymptotic distributions, of the maximum likelihood estimators of the parameters of the corresponding distributions, and of the respective system reliabilities are presented. The bootstrap estimate and confidence interval are also obtained. The procedures are illustrated based on certain datasets.

This paper proposes a new method to identify local damages in frame structures based on approximate Metropolis–Hastings (AMH) algorithm and statistical moment. By analyzing the sensitivity of different statistical moment-based damage indices, the fusion index of fourth-order displacement moment and eighth-order acceleration moment is selected. Then the local damages in frame structures are primarily evaluated by AMH algorithm, where Gibbs sampling is adopted. Finally, the uncertainty of identified local damages is analyzed by using probability density evolution method (PDEM). Numerical simulations have been conducted to compare the proposed method with other similar damage detection methods, showing that the proposed method is more time-saving due to the involvement of Gibbs sampling and more accurate in assessing the damage severity. Experimental study of a 12-story benchmark frame testing has also been carried out, further validating the effectiveness of the proposed method.

Vibration-based global damage detection based on updating of finite element (FE) model by targeting the modal measurements is a significant area of interest in structural health monitoring (SHM). In a typical modal testing setup, the measured mode shapes have missing components against various degrees of freedom (DOFs) due to the limitation in the number of sensors available. In this context, a novel Gibbs sampling approach is proposed for updating of FE model incorporating model reduction (MR) to facilitate the global-level detection of structural damages from incomplete modal measurements. In addition to the ease with similar sizes of analytical and experimental mode shapes, the proposed Gibbs sampling approach (for updating the reduced order FE model in the Bayesian framework) has some important advantages like: (A) no need for consideration of system mode shapes as parameters (unlike needed in the typical Gibbs sampling approach) thereby having a significant reduction in the number of parameters, (B) non-requirement of mode matching with consequent reduction in computation time to a significant extent. A generalized formulation is presented in this work providing the scope for incorporating measurements from multiple sensor setups. Moreover, formulations are adapted to incorporate multiple sets of data/measurements from each setup targeting the epistemic uncertainty. Finally, validation is carried out with both numerical (truss structure and building structure) and experimental (laboratory building structure) exercises in comparison with the typical Gibbs sampling approach having a full-sized model. The proposed approach is observed to be evolved as a computationally efficient technique with satisfactory performance in FE model updating and global damage detection.

The identification of multiple change points is a problem shared by many subject areas, including disease and criminality mapping, medical diagnosis, industrial control, and finance. An algorithm based on the Product Partition Model (PPM) is developed to solve the multiple change point identification problem in Poisson data sequences. In order to address the PPM, a simple and easy way to implement Gibbs sampling scheme is derived. A sensitivity analysis is performed, for different prior specifications. The algorithm is then applied to the analysis of a real data sequence. The results show that the method is quite effective and provides useful inferences.

In this paper, we study Bayesian analysis of nonlinear hierarchical mixture models with a finite but unknown number of components. Our approach is based on Markov chain Monte Carlo (MCMC) methods. One of the applications of our method is directed to the clustering problem in gene expression analysis. From a mathematical and statistical point of view, we discuss the following topics: theoretical and practical convergence problems of the MCMC method; determination of the number of components in the mixture; and computational problems associated with likelihood calculations. In the existing literature, these problems have mainly been addressed in the linear case. One of the main contributions of this paper is developing a method for the nonlinear case. Our approach is based on a combination of methods including Gibbs sampling, random permutation sampling, birth-death MCMC, and Kullback–Leibler distance.

Most of the associated single nucleotide polymorphisms (SNPs) for genome wide association studies (GWAS) explain very little proportion of phenotypic variance in outbred populations. One reason is; large number of markers raises the problem of multiple hypothesis testing correction using conservative statistical tests in single marker models. Admixture mapping could be used as alternative model to detect the genes associated with quantitative traits by less number of ancestry informative markers. Ancestral genotypes of founder populations were available for the F₂ mice dataset for growth related traits. The objectives of this study were (1) to detect genomic signals by admixture mapping for growth related traits by ancestry informative markers and ancestral genotypes (2) to detect genomic signals for growth related traits by Bayes C(π) model and compare results with those obtained by use of admixture mapping. Bayes C(π) model detected more SNPs that has high ancestry informative markers. But due to stringent significance tests and small SNPs effects admixture model did not detect the same SNPs in Bayes C(π). As was expected higher ancestral informative markers lead to higher Z values in admixture model with a little variation. Admixture model could incorporate and use ancestral genomic information.

This paper deals with Bayesian estimation of parameters in the proportional hazards model of random censorship for the Weibull distribution under different loss functions. We consider both the informative and noninformative priors on the model parameters to obtain the Bayes estimates using Gibbs sampling scheme. Maximum likelihood estimates are also obtained for comparison purposes. A simulation study is carried out to observe the behavior of the proposed estimators for different sample sizes and for different censoring parameters. One real data analysis is performed for illustration.

The maximum likelihood estimation of correlation matrix under order restrictions among correlations is treated. Two maximization process:(A) maximization with respect to correlation matrix and (B) maximization with respect to variance are considered. For the maximization process (A), we generate uniformly distributed random correlation matrices on the hypothesis space by Gibbs sampling. In the maximization process (B), we show that the maximum point is the fixed point of the iterative application of a certain non-linear function and the convergence of the process (B) is proved. A simulation result is given which compares the relative errors of the m.l.e. and other competitors.

We provide a tutorial on learning and inference in hidden Markov models in the context of the recent literature on Bayesian networks. This perspective makes it possible to consider novel generalizations of hidden Markov models with multiple hidden state variables, multiscale representations, and mixed discrete and continuous variables. Although exact inference in these generalizations is usually intractable, one can use approximate inference algorithms such as Markov chain sampling and variational methods. We describe how such methods are applied to these generalized hidden Markov models. We conclude this review with a discussion of Bayesian methods for model selection in generalized HMMs.