Search Results (91)

Markov chain Monte Carlo (MCMC) methods are fundamental to Bayesian inference but are often computationally prohibitive for large datasets, as the full likelihood must be evaluated at each iteration. Subsampling-based approximate Metropolis–Hastings (MH) algorithms offer a popular alternative, trading a manageable bias for a significant reduction in per-iteration cost. While this bias–variance trade-off is empirically understood, a formal theoretical framework for its optimization has been lacking. Our work establishes such a framework by bounding the mean squared error (MSE) as a function of the subsample size (m), the data size (n), and the number of epochs (E). This analysis reveals two optimal asymptotic scaling laws: the optimal subsample size is $m^{★}=O\left(E^{1/2}\right)$, leading to a minimal MSE that scales as ${\mathrm{MSE}}^{★}=O\left(E^{-1/2}\right)$. Furthermore, leveraging the large-sample asymptotic properties of the posterior, we show that when augmented with a control variate, the approximate MH algorithm can be asymptotically more efficient than the standard MH method under ideal conditions. Experimentally, we first validate the two optimal asymptotic scaling laws. We then use Bayesian logistic regression and Softmax classification models to highlight a key difference in convergence behavior: the exact algorithm starts with a high MSE that gradually decreases as the number of epochs increases. In contrast, the approximate algorithm with a practical control variate maintains a consistently low MSE that is largely insensitive to the number of epochs. Full article

This paper presents a comprehensive study on the estimation of multicomponent stress–strength reliability under progressively censored data, assuming the inverse Rayleigh distribution. Both maximum likelihood estimation and Bayesian estimation methods are considered. The loss function and prior distribution play crucial roles in Bayesian inference. Therefore, Bayes estimators of the unknown model parameters are obtained under symmetric (squared error loss function) and asymmetric (linear exponential and general entropy) loss functions using gamma priors. Lindley and MCMC approximation methods are used for Bayesian calculations. Additionally, asymptotic confidence intervals based on maximum likelihood estimators and Bayesian credible intervals constructed via Markov Chain Monte Carlo methods are presented. An extensive Monte Carlo simulation study compares the efficiencies of classical and Bayesian estimators, revealing that Bayesian estimators outperform classical ones. Finally, a real-life data example is provided to illustrate the practical applicability of the proposed methods. Full article

The present research proposes a methodology for portfolio construction that integrates the Black–Litterman model with expected returns generated through simulations under dynamic Capital Asset Pricing Model (CAPM) with conditional betas, estimated via Approximate Bayesian Computation Markov Chain Monte Carlo (ABC-MCMC). Bayesian estimation enables the incorporation of volatility regimes and the adjustment of each asset’s sensitivity to the market, thereby delivering expected returns that more accurately reflect the structural state of the assets compared to historical methods. This strategy is applied to the United States stock market, and the results suggest that the Black–Litterman portfolio performs competitively against portfolios optimised using the classic Markowitz model, even maintaining the same fixed weights throughout the month. Specifically, it has been demonstrated to outperform the minimum variance portfolio with regard to cumulative return and attains a Sharpe ratio that approaches the Markowitz maximum Sharpe portfolio, although it does so with a distinct and more concentrated asset allocation. It has been observed that, while the maximum return portfolio attains the highest absolute profit, it does so at the expense of significantly higher volatility. Full article

Using a Bayesian framework to invert earthquake source parameters from multi-source geodetic data has become an important research direction. To address the issue of Markov Chain Monte Carlo (MCMC) algorithms getting stuck in local optima during nonlinear inversion of fault geometric parameters, which is often caused by improperly set parameter bounds or large deviations in the initial values, this study proposes two strategies: ‘CFI (Converge First, Then Interval)’ and ‘IVI (Interval Value Iteration)’. Tests with 12 different experimental setups show that both strategies can prevent the chain from getting trapped in local optima. Among them, the ‘IVI’ strategy, when used with MCMC algorithms where the step size follows a normal distribution, can also significantly reduce the root-mean-square error. To verify its applicability, the ‘IVI’ strategy was applied to the Bayesian inversion of the 2022 Menyuan Mw6.6 earthquake. The results show that the inverted values for fault depth, strike, dip, and rake angles are closer to the GCMT results, with ascending and descending track fitting residuals of 2.71 cm and 2.64 cm, respectively. The conclusion of this paper is to recommend the ‘IVI’ strategy when the range of source parameters is unclear. If the approximate range of parameters is known, the ‘CFI’ strategy can be applied. The original interval constraint method is recommended when the parameter bounds are fully determinable and a reliable initial model of seismic source parameters is obtainable. Full article

We present empirical evidence on the sensitivity of housing demand in Chile to changes in mortgage interest rates, focusing on units priced between CLF 2000 and 4000 (approximately USD 80,000 to 160,000). This sector, which comprises nearly two-thirds of the country’s housing supply, has experienced a significant decline in sales since 2021. Given its size and responsiveness, it represents a key target for policy measures aimed at reactivating the Chilean real estate market, such as demand-side subsidies for middle-income households. Using impulse response functions derived from vector autoregressive (VAR) and semi-structural models estimated via Bayesian methods with Markov Chain Monte Carlo (MCMC) simulations, we find that a 100-basis-point increase in mortgage rates leads to an average annual decline of 18% in housing sales during the first quarter after the shock. This effect results in a cumulative decline of approximately 57% by the end of the first year. A comparable reduction in mortgage rates yields a symmetrical response. Finally, we offer a linear extrapolation of potential impacts under a hypothetical 200-basis-point decrease in mortgage rates. Full article

The Hubble tension, a persistent discrepancy between early and late Universe measurements of $H_0$, poses a significant challenge to the standard cosmological model. In this work, we present a new Bayesian hierarchical framework designed to meticulously decompose this observed tension into its constituent parts: standard measurement errors, information loss arising from parameter-space projection, and genuine physical tension. Our approach, employing Fisher matrix analysis with MCMC-estimated loss coefficients and explicitly modeling information loss via variance inflation factors ($\lambda$), is particularly important in high-precision analysis where even seemingly small information losses can impact conclusions. We find that the real tension component ($T_{real}$) has a mean value of 5.94 km/s/Mpc (95% CI: [3.32, 8.64] km/s/Mpc). Quantitatively, approximately 78% of the observed tension variance is attributed to real tension, 13% to measurement error, and 9% to information loss. Despite this, our decomposition indicates that the observed ∼$6.39\sigma$ discrepancy is predominantly a real physical phenomenon, with real tension contributing ∼$5.64\sigma$. Our findings strongly suggest that the Hubble tension is robust and probably points toward new physics beyond the ΛCDM model. Full article

Industrial systems often rely on specialized redundant systems to enhance reliability and prevent unexpected failures. This study introduces a novel three-parameter model, the Poisson–Weibull distribution (PWD), and discovers its various key properties. The primary focus of the study is to develop stress–strength (SS) model based on this newly developed distribution. Parameter estimation for both the PWD and SS models is carried out using maximum likelihood estimation (MLE) and Bayesian estimation techniques. Given the complexity of the proposed distribution, numerical approximation techniques are employed to obtain reliable parameter estimates. A comprehensive simulation study employing the Monte Carlo simulation (MCS) and Markov Chain Monte Carlo (MCMC) examines the behavior of the PWD and SS model parameters under various scenarios. The development of the SS model enhances understanding of the PWD’s dynamics while providing practical insights into its real-life applications and limitations. The effectiveness of the proposed distribution and the SS reliability measure is established through applications to real-life data sets. Full article

The stability of modern power systems has become critically dependent on precise inertia estimation of synchronous generators, particularly as renewable energy integration fundamentally transforms grid dynamics. Increasing penetration of converter-interfaced renewable resources reduces system inertia, heightening the grid’s susceptibility to transient disturbances and creating significant technical challenges in maintaining operational reliability. This paper addresses these challenges through a novel Bayesian inference framework that synergistically integrates PMU data with an advanced MCMC sampling technique, specifically employing the Affine-Invariant Ensemble Sampler. The proposed methodology establishes a probabilistic estimation paradigm that systematically combines prior engineering knowledge with real-time measurements, while the Affine-Invariant Ensemble Sampler mechanism overcomes high-dimensional computational barriers through its unique ensemble-based exploration strategy featuring stretch moves and parallel walker coordination. The framework’s ability to provide full posterior distributions of inertia parameters, rather than single-point estimates, helps for stability assessment in renewable-dominated grids. Simulation results on the IEEE 39-bus and 68-bus benchmark systems validate the effectiveness and scalability of the proposed method, with inertia estimation errors consistently maintained below $1\%$ across all generators. Moreover, the parallelized implementation of the algorithm significantly outperforms the conventional M-H method in computational efficiency. Specifically, the proposed approach reduces execution time by approximately $52\%$ in the 39-bus system and by $57\%$ in the 68-bus system, demonstrating its suitability for real-time and large-scale power system applications. Full article

Spatial data not only enables smart cities to visualize, analyze, and interpret data related to location and space, but also helps departments make more informed decisions. We apply a Bayesian quantile regression (BQR) of the partially linear varying coefficient spatial autoregressive (PLVCSAR) model for spatial data to improve the prediction of performance. It can be used to capture the response of covariates to linear and nonlinear effects at different quantile points. Through an approximation of the nonparametric functions with free-knot splines, we develop a Bayesian sampling approach that can be applied by the Markov chain Monte Carlo (MCMC) approach and design an efficient Metropolis–Hastings within the Gibbs sampling algorithm to explore the joint posterior distributions. Computational efficiency is achieved through a modified reversible-jump MCMC algorithm incorporating adaptive movement steps to accelerate chain convergence. The simulation results demonstrate that our estimator exhibits robustness to alternative spatial weight matrices and outperforms both quantile regression (QR) and instrumental variable quantile regression (IVQR) in a finite sample at different quantiles. The effectiveness of the proposed model and estimation method is demonstrated by the use of real data from the Boston median house price. Full article

This article presents maximum likelihood and Bayesian estimates for the parameters, reliability function, and hazard function of the Gumbel Type-II distribution using a unified hybrid censored sample. Bayesian estimates are derived under three loss functions: squared error, LINEX, and generalized entropy. The parameters are assumed to follow independent gamma prior distributions. Since closed-form solutions are not available, the MCMC approximation method is used to obtain the Bayesian estimates. The highest posterior density credible intervals for the model parameters are computed using importance sampling. Additionally, approximate confidence intervals are constructed based on the normal approximation to the maximum likelihood estimates. To derive asymptotic confidence intervals for the reliability and hazard functions, their variances are estimated using the delta method. A numerical study compares the proposed estimators in terms of their average values and mean squared error using Monte Carlo simulations. Finally, a real dataset is analyzed to illustrate the proposed estimation methods. Full article

In recent years, there has been an increasing interest in the application of progressive censoring as a means to reduce both cost and experiment duration. In the absence of explanatory variables, the present study employs a statistical inference approach for the inverse Weibull distribution, using a progressive type II censoring strategy with two independent samples. The article expounds on the maximum likelihood estimation method, utilizing the Fisher information matrix to derive approximate confidence intervals. Moreover, interval estimations are computed by the bootstrap method. We explore the application of Bayesian methods for estimating model parameters under both the squared error and LINEX loss functions. The Bayesian estimates and corresponding credible intervals are calculated via Markov chain Monte Carlo (MCMC). Finally, comprehensive simulation studies and real data analysis are carried out to validate the precision of the proposed estimation methods. Full article

Silicone structural glazing (SSG) sealants are crucial sealing materials in modern building curtain walls, whose performance degradation may lead to functional and safety issues, posing significant challenges to building safety maintenance. This study comprehensively investigated the effects of temperature, humidity, stress, and ultraviolet (UV) irradiance on the durability of SSG sealants through multi-gradient matrix aging tests, revealing the influence patterns of these four aging factors on tensile bond strength (TBS). Based on aging test data and degradation patterns, a novel degradation model for TBS aging was established by incorporating all four aging factors as variables, enabling the model to reflect their combined effects on TBS degradation. The unknown parameters in the model were calculated using the Markov chain Monte Carlo (MCMC) algorithm and validated against experimental data. A recursive algorithm was developed to predict TBS degradation under actual service conditions based on the degradation model and environmental records, with verification through outdoor aging tests. This study established a service life prediction methodology that combines the degradation model with environmental data through recursive computation and standard-specified strength limits. The results demonstrate that increasing temperature, humidity, stress, and UV irradiation accelerates TBS changes, with influence intensity ranking as UV irradiation > temperature > humidity > stress. Synergistic effects exist among all four factors, where UV irradiation shows the most significant coupling effect by amplifying other factors’ combined impacts, while UV’s primary influence manifests through such synergies rather than independent action. Among temperature, humidity, and stress combined effects, temperature contributes approximately 50%, temperature–humidity interaction about 35%, with temperature-related terms collectively accounting for 90%. The degradation model calculation results show excellent agreement with experimental data (R² > 0.9, MAE = 0.019 MPa, RMSE = 0.0245 MPa). The characteristic TBS minimum value considering material discreteness and strength assurance rate serves as a reliable criterion for service life evaluation. The proposed prediction method provides essential theoretical and methodological foundations for ensuring long-term safety and maintenance strategies for glass curtain walls. Full article

This paper presents a Bayesian-based finite element model updating method that integrates static displacement measurements and dynamic modal data to simultaneously identify structural mass and stiffness parameters. By leveraging Bayesian inference, a posterior probability density function (PDF) is constructed by integrating static displacement and modal parameters, thereby effectively decoupling the identification of structural mass and stiffness. The Delayed Rejection Adaptive Metropolis (DRAM) Markov Chain Monte Carlo (MCMC) sampling algorithm is utilized to derive the posterior distributions of the updated parameters. To mitigate the computational burden associated with repetitive finite element (FE) analyses during large-scale MCMC sampling, a Kriging surrogate model is employed to efficiently approximate the time-consuming FE simulations. Numerical examples involving a cantilever beam and an actual concrete three-span single-box girder bridge illustrate that the proposed method accurately identifies simultaneous variations in mass and stiffness at multiple structural locations, effectively addressing parameter coupling and misidentification issues encountered when using either static or dynamic data alone. Moreover, the Kriging surrogate significantly improves computational efficiency. Experimental validation on an aluminum alloy cantilever beam further corroborates the effectiveness and practical applicability of the proposed method. Full article

Bayesian methods have revolutionised hydrological modelling by providing a framework for managing uncertainty, improving model calibration, and enabling more accurate predictions. This paper reviews the evolution of Bayesian methods in hydrology, from their initial applications in flood-frequency analysis to their current use in streamflow forecasting, flood risk assessment, and climate-change adaptation. It discusses the development of key Bayesian techniques, such as Markov Chain Monte Carlo (MCMC) methods, hierarchical models, and approximate Bayesian computation (ABC), and their integration with remote sensing and big data analytics. The paper also presents simulated examples demonstrating the application of Bayesian methods to flood, drought, and rainfall data, showcasing the potential of these methods to inform water-resource management, flood risk mitigation, and drought prediction. The future of Bayesian hydrology lies in expanding the use of machine learning, improving computational efficiency, and integrating large-scale datasets from remote sensing. This review serves as a resource for hydrologists seeking to understand the evolution and future potential of Bayesian methods in addressing complex hydrological challenges. Full article

The conventional diagnostic techniques for ethylene cracker furnace tube coking rely on manual expertise, offline analysis and on-site inspection. However, these methods have inherent limitations, including prolonged inspection times, low accuracy and poor real-time performance. This makes it challenging to meet the requirements of chemical production. The necessity for high efficiency, high reliability and high safety, coupled with the inherent complexity of the production process, results in data that is characterized by multimodal, nonlinear, non-Gaussian and strong noise. This renders the traditional data processing and analysis methods ineffective. In order to address these issues, this paper puts forth a novel soft measurement approach, namely the ‘Mixed Student’s t-distribution regression soft measurement model based on Variational Inference (VI) and Markov Chain Monte Carlo (MCMC)’. The initial variational distribution is selected during the initialization step of VI. Subsequently, VI is employed to iteratively refine the distribution in order to more closely approximate the true posterior distribution. Subsequently, the outcomes of VI are employed to initiate the MCMC, which facilitates the placement of the iterative starting point of the MCMC in a region that more closely approximates the true posterior distribution. This approach allows the convergence process of MCMC to be accelerated, thereby enabling a more rapid approach to the true posterior distribution. The model integrates the efficiency of VI with the accuracy of the MCMC, thereby enhancing the precision of the posterior distribution approximation while preserving computational efficiency. The experimental results demonstrate that the model exhibits enhanced accuracy and robustness in the diagnosis of ethylene cracker tube coking compared to the conventional Partial Least Squares Regression (PLSR), Gaussian Process Regression (GPR), Gaussian Mixture Regression (GMR), Bayesian Student’s T-Distribution Mixture Regression (STMR) and Semi-supervised Bayesian T-Distribution Mixture Regression (SsSMM). This method provides a scientific basis for optimizing and maintaining the ethylene cracker, enhancing its production efficiency and reliability, and effectively addressing the multimodal, non-Gaussian distribution and uncertainty of the coking data of the ethylene cracker furnace tube. Full article