This paper proposes a Bayesian MCMC-INLA algorithm specifically designed for both unidimensional and multidimensional logistic graded response models (LGRMs). The algorithm incorporates a computationally efficient data augmentation approach by introducing Pólya-Gamma variables and latent variables, thereby addressing the limitations of traditional Bayesian MCMC methods in handling item response theory (IRT) models with logistic link functions. By integrating the advanced and efficient integrated nested Laplace approximation (INLA) framework, the MCMC-INLA algorithm achieves both high computational efficiency and estimation accuracy. The paper provides detailed derivations of the posterior and conditional distributions for IRT models, outlines the incorporation of Pólya-Gamma and latent variables within the Gibbs sampling procedure, and presents the implementation of the MCMC-INLA algorithm for both unidimensional and multidimensional cases. The performance of the proposed algorithm is evaluated through extensive simulation studies and an empirical application to the IPIP-NEO dataset. Potential extensions of the MCMC-INLA framework to other IRT models are also discussed.

Multidimensional degradation processes commonly arise in complex engineering systems such as aerospace equipment, military devices, and new-energy vehicles. These systems exhibit degradation behaviors characterized by multidimensionality, dependence, and heterogeneity. To effectively capture these features, this article proposes a parameter-dependent multivariate Wiener process degradation model that simultaneously accounts for correlations among multiple degradation characteristics, dependencies between degradation rate and volatility, and individual heterogeneity within subsystems. A hierarchical Bayesian inference framework based on Gibbs sampling is developed for joint parameter estimation and uncertainty quantification. To address estimation bias issues in high-dimensional covariance structures, a hierarchical inverse Wishart prior is introduced, improving the accuracy of covariance estimation, particularly under small-variance conditions. Furthermore, a Monte Carlo-based reliability estimation scheme is proposed to approximate the first-passage time distribution of multidimensional systems. Extensive simulation studies demonstrate the superior estimation accuracy and credible interval coverage of the proposed method compared to traditional priors. Finally, applications to real-world engineering degradation data validate the model's practical effectiveness, showing that neglecting the dependence between degradation rate and volatility can lead to substantial bias in reliability assessments and maintenance decisions.

Step-stress accelerated degradation testing (SSADT) plays a critical role in evaluating the reliability of highperformance industrial products under harsh conditions, where performance deterioration is not significant under normal operating conditions. However, existing Bayesian inference methods for SSADT models face significant challenges due to computational inefficiency, particularly in achieving convergence and handling complex stochastic processes. These limitations hinder practical applications where rapid and precise reliability assessment is essential. To address this, we propose a novel iterative integrated nested Laplace approximation framework combined with a fixed-point iteration technique. By reformulating the Wiener-process-based SSADT model into a latent Gaussian model via Taylor linearization, our approach leverages quadratic polynomial approximation and expansion-and-contraction strategies to optimize computational efficiency. Simulation studies demonstrate that the proposed method achieves comparable accuracy to traditional Bayesian methods like Gibbs sampling while significantly reducing computational costs, even for moderate sample sizes. Additionally, empirical validation using two real-world datasets confirms its applicability and effectiveness in practical reliability analysis.

Multidimensional graded response models (MGRMs) are widely used for analyzing ordinal questionnaire data in psychological and educational assessments. A central challenge in applying these models is determining the number of latent dimensions. Conventional approaches usually fit multiple fixed-dimensional models and select among them using post-hoc criteria such as AIC, BIC, or cross-validation, which can be computationally demanding and ignore uncertainty in dimensionality during estimation. We develop an adaptive Bayesian dimension selection framework for probit MGRMs. Building on the cumulative shrinkage process, we assign a cumulative ordered spike-and-slab (COSS) prior to the column-specific variances of the item loading matrix. This prior induces increasing shrinkage across latent dimensions, allowing redundant dimensions to be shrunk toward zero while preserving flexibility for active dimensions. Albert–Chib latent response augmentation is used to handle the ordinal probit likelihood, yielding conditionally Gaussian updates for item loadings and latent traits. These updates are combined with Gibbs updates for threshold and shrinkage parameters in an efficient adaptive sampler. Simulation studies evaluate the proposed method in terms of dimension recovery, parameter estimation accuracy, and computational efficiency, with comparisons to conventional fixed-dimensional estimation and model selection procedures. The results show that the proposed approach accurately recovers the latent structure while avoiding repeated model fitting over multiple candidate dimensions. We further illustrate the method using real psychological assessment data, demonstrating its practical utility for uncovering interpretable latent structures in ordinal item responses.

The integrated nested Laplace approximation (INLA) algorithm provides a computationally efficient approach for approximate Bayesian inference, overcoming the limitations of traditional Markov chain Monte Carlo (MCMC) methods. This paper reviews INLA algorithm and provides a systematic review of six key books that explore the theoretical foundations, practical implementations, and diverse applications of INLA. These six books cover spatial and spatio-temporal modelling, general Bayesian inference, SPDE-based spatial analysis, geospatial health data, regression modelling, and dynamic time series. In addition, these books highlight the versatility of INLA method in handling complex models while maintaining high computational efficiency. This paper begins with an introduction to the INLA method and algorithm, followed by a systematic review of six key publications in the field.

Modeling the joint degradation of two or more performance characteristics (PCs) is essential for accurately assessing the reliability of complex systems. However, a significant challenge remains when these PCs exhibit heterogeneous behaviors, specifically when the degradation paths comprise a mix of Gaussian and heavy-tailed characteristics. To address this, we propose a novel bivariate Wiener degradation model incorporating a partially random scale structure. This hybrid model maintains a Gaussian path for one PC while introducing an inverse-gamma-distributed random scale effect for the other, yielding a flexible Student-t process that captures tail heaviness and accounts for the correlation between the two processes. For statistical inference, we develop an expectation-maximization algorithm and a Bayesian method for parameter estimation, with a percentile bootstrap approach for constructing confidence intervals. The effectiveness of the proposed model and inference methods is validated through comprehensive simulation studies and an application to a permanent magnet brake degradation dataset.

A major challenge in spatial transcriptomics (ST) is resolving cellular composition, especially in technologies lacking single-cell resolution. The mixture of transcriptional signals within spatial spots complicates deconvolution and downstream analyses. To uncover the spatial heterogeneity of tissues, we introduce SvdRFCTD, a reference-free spatial transcriptomics deconvolution method, which estimates the cell type proportions at each spot on the tissue. To fully capture the heterogeneity in the ST data, we combine SvdRFCTD with a Bayesian hierarchical negative binomial model with spatial effects incorporated in both the mean and dispersion of the gene expression, which is used to explicitly model the generative mechanism of cell type proportions. By integrating spatial information and leveraging marker gene information, SvdRFCTD accurately estimates cell type proportions and uncovers complex spatial patterns. We demonstrate the ability of SvdRFCTD to identify cell types on simulated datasets. By applying SvdRFCTD to mouse brain and human pancreatic ductal adenocarcinomas datasets, we observe significant cellular heterogeneity within the tissue sections and successfully identify regions with high proportions of aggregated cell types, along with the spatial relationships between different cell types.

Network models are increasingly vital in psychometrics for analyzing relational data, which are often accompanied by high-dimensional node attributes. Joint latent space models (JLSM) provide an elegant framework for integrating these data sources by assuming a shared underlying latent representation; however, a persistent methodological challenge is determining the dimension of the latent space, as existing methods typically require pre-specification or rely on computationally intensive post-hoc procedures. The key innovation of this work is a cumulative ordered spike-and-slab (COSS) prior, which we incorporate within a Bayesian joint latent space modeling framework. This prior enables the latent dimension to be inferred automatically and simultaneously with all model parameters. We develop an efficient Markov Chain Monte Carlo (MCMC) algorithm for posterior computation. Theoretically, we establish that the posterior distribution concentrates on the true latent dimension and that parameter estimates achieve Hellinger consistency at a near-optimal rate that adapts to the unknown dimensionality. Through extensive simulations and three real-data applications, we demonstrate the method's superior performance in both dimension recovery and parameter estimation. Our work offers a principled, computationally efficient, and theoretically grounded solution for adaptive dimension selection in psychometric network models.

In this paper, we propose a Bayesian PG-INLA algorithm which is tailored to both one-dimensional and multidimensional 2-PL IRT models. The proposed PG-INLA algorithm utilizes a computationally efficient data augmentation strategy via the Pólya-Gamma variables, which can avoid low computational efficiency of traditioanl Bayesian MCMC algorithms for IRT models with a logistic link function. Meanwhile, combined with the advanced and fast INLA algorithm, the PG-INLA algorithm is both accurate and computationally efficient. We provide details on the derivation of posterior and conditional distributions of IRT models, the method of introducing the Pólya-Gamma variable into Gibbs sampling, and the implementation of the PG-INLA algorithm for both one-dimensional and multidimensional cases. Through simulation studies and an application to the data analysis of the IPIP-NEO personality inventory, we assess the performance of the PG-INLA algorithm. Extensions of the proposed PG-INLA algorithm to other IRT models are also discussed.

Remaining Useful Life (RUL) is one of the most important indicators to detect a component failure. RUL can be predicted by historical data by adopting a model-based method. The stochastic process models have become the most popular way to model degradation data for high-quality products, such as the Wiener process, gamma process and inverse Gaussian process. However, this leads to poor reliability assessment if the model is misspecified. Application of the Tweedie exponential dispersion (TED) process, including the above-mentioned classical stochastic processes as special cases, transforms the model selection problem into a parameter estimation problem dexterously. In this paper, we propose a TED process with random drifts for degradation data and a TED process with random drifts and covariates for accelerated degradation data. A hierarchical Bayesian method is adopted to estimate the parameters of the proposed models. We also derive the failure-time distribution and the remaining useful life distribution for the proposed models. The simulation study shows that the proposed model outperforms the wrongly specified models. Two illustrative examples demonstrate the performance of the proposed TED process with random drifts and the TED process with random drifts and covariates.

Two-phase degradation is a prevalent degradation mechanism observed in modern systems, typically characterized by a change in the degradation rate or trend of a system’s performance at a specific time point. Ignoring this change in degradation models can lead to considerable biases in predicting the remaining useful life (RUL) of the system, and potentially leading to inappropriate condition-based maintenance decisions. To address this issue, we propose a novel two-phase degradation model based on a reparameterized inverse Gaussian process. The model considers variations in both change points and model parameters among different systems to account for subject-to-subject heterogeneity. The unknown parameters are estimated using both maximum likelihood and Bayesian approaches. Additionally, we propose an adaptive replacement policy based on the distribution of RUL. By sequentially obtaining new degradation data, we dynamically update the estimation of model parameters and of the RUL distribution, allowing for adaptive replacement policies. A simulation study is conducted to assess the performance of our methodologies. Finally, a Lithium-ion battery example is provided to validate the proposed model and adaptive replacement policy. Technical details and additional results of case study are available as online supplementary materials.

In this paper, we propose a novel Gibbs-INLA algorithm for the Bayesian inference of graded response models with ordinal response based on multidimensional item response theory. With the combination of the Gibbs sampling and the integrated nested Laplace approximation (INLA), the new framework avoids the cumbersome tuning which is inevitable in classical Markov chain Monte Carlo (MCMC) algorithm, and has low computing memory, high computational efficiency with much fewer iterations, and still achieve higher estimation accuracy. Therefore, it has the ability to handle large amount of multidimensional response data with different item responses. Simulation studies are conducted to compare with the Metroplis-Hastings Robbins-Monro (MH-RM) algorithm and an application to the study of the IPIP-NEO personality inventory data is given to assess the performance of the new algorithm. Extensions of the proposed algorithm for application on more complicated models and different data types are also discussed.

In the field of reliability engineering, the gamma process plays an important role in modeling degradation processes. However, extracting lifetime information from product degradation observations has long been suffering from both ineffective modeling techniques and inefficient statistical inference methods. To overcome these challenges, we propose a reparameterized gamma process with random effects in this article. Compared with the classical gamma process, the proposed model has a more intuitive physical interpretation. In addition, statistical inference for the model can be readily done through the variational Bayesian algorithm. Combining with the Gauss–Hermite quadrature and the Laplace approximation, the algorithm yields closed-form variational posteriors for the proposed model. Its superiority over two other inference methods (expectation maximization and Monte Carlo Markov Chain) in terms of computational efficiency and estimation accuracy is demonstrated by simulation.

In industry, many highly reliable products possess multiple performance characteristics (PCs) and they typically degrade simultaneously. When such PCs are governed by a common failure mechanism or influenced by a shared operating environmental condition, interdependence between these PCs arises. To model such dependence, this article proposes a novel multivariate reparameterized inverse Gaussian (rIG) process model. It utilizes an additive structure; that is, the degradation of each marginal PC is considered as the result of the sum of two independent rIG processes, with one capturing the shared common effects across all PCs and the other describing the intrinsic randomness specific to that PC. The model has some nice statistical properties, and the system lifetime distribution can be conveniently approximated. An expectation-maximization algorithm is proposed for estimating the model parameters, and a parametric bootstrap method is designed to derive the confidence intervals. Comprehensive numerical simulations are conducted to validate the performance of the inference method. Two case studies are thoroughly investigated to demonstrate the applicability of the proposed methodology. Supplementary materials for this article are available online.

The accelerated failure time mixture cure (AFTMC) model is widely used for survival data when a portion of patients can be cured. In this paper, a Bayesian semiparametric method is proposed to obtain the estimation of parameters and density distribution for both the cure probability and the survival distribution of the uncured patients in the AFTMC model. Specifically, the baseline error distribution of the uncured patients is nonparametrically modeled by a mixture of Dirichlet process. Based on the stick-breaking formulation of the Dirichlet process, the techniques of retrospective and slice sampling, an efficient and easy-to-implement Gibbs sampler is developed for the posterior calculation. The proposed approach can be easily implemented in commonly used statistical softwares, and its performance is comparable to fully parametric method via comprehensive simulation studies. Besides, the proposed approach is adopted to the analysis of a colorectal cancer clinical trial data.

As the development of the technology and material science,products become highly reliable with long lifetimes.Consequently,the traditional life tests are gradually replaced by the degradation tests.In the degradation tests,the quality characteristic(QC)which degrades over time,can reflect the reliability status of the product.Some of the degradation paths of the QC are usually monotonically increasing and non-smooth function of time.In this paper,motivated by the degradation data from the tests of N-channel power metal oxide semiconductor field-effect transistor(MOSFET),we propose a two-phase gamma process under hierarchical Bayesian framework to model the monotonic and non-smooth degradation data.A simulation study is performed to verify the utility of the proposed model under three scenarios.An illustrative anti-radiation performance case study is analyzed to show the applicable power of the proposed model.

In this paper, we proposed an interval degradation model to improve the reliability of the classical single point degradation model. The interval degradation model is very flexible when model parameters follows different distributions. Twenty-five types of interval Gamma degradation models are considered and discussed under different conditions. The reliabilities of interval Gamma degradation models are obtained. The Monte Carlo method has been studied to compute the reliability and lifetime of interval Gamma degradation model. The numerical examples are conducted to compare the interval degradation model with the classical single point degradation model. Simulation results reveal that the performance of reliability and mean lifetime of interval Gamma degradation model are much better than those of the single Gamma degradation model. Finally, we applied our model to a real data example and demonstrated the effectiveness and feasibility of the interval Gamma degradation model.

Gradient descent (GD) algorithm is the widely used optimisation method in training machine learning and deep learning models. In this paper, based on GD, Polyak's momentum (PM), and Nesterov accelerated gradient (NAG), we give the convergence of the algorithms from an initial value to the optimal value of an objective function in simple quadratic form. Based on the convergence property of the quadratic function, two sister sequences of NAG's iteration and parallel tangent methods in neural networks, the three-step accelerated gradient (TAG) algorithm is proposed, which has three sequences other than two sister sequences. To illustrate the performance of this algorithm, we compare the proposed algorithm with the three other algorithms in quadratic function, high-dimensional quadratic functions, and nonquadratic function. Then we consider to combine the TAG algorithm to the backpropagation algorithm and the stochastic gradient descent algorithm in deep learning. For conveniently facilitate the proposed algorithms, we rewite the R package 'neuralnet' and extend it to 'supneuralnet'. All kinds of deep learning algorithms in this paper are included in 'supneuralnet' package. Finally, we show our algorithms are superior to other algorithms in four case studies.

In this paper, a new test statistic based on the weighted Frobenius norm of covariance matrices is proposed to test the homogeneity of multi-group population covariance matrices. The asymptotic distributions of the proposed test under the null and the alternative hypotheses are derived, respectively. Simulation results show that the proposed test procedure tends to outperform some existing test procedures.