Bayesian Estimation Research Articles

Inference Based on Sequential Order Statistics with Lindley Distribution

In this paper, we consider the problem of estimation of parameters of Lindley distribution based on sequential order statistics under conditional proportional hazard rate model. Maximum Likelihood and Bayesian estimation techniques are applied for estimating the unknown parameters. Markov Chain Monte Carlo method is implemented for obtaining the Bayes estimators. An intensive simulation study is carried out to assess the performance of different estimators discussed in this paper. A real data analysis is also performed for illustrative purposes.

Bayesian parameter estimation using truncated normal distributions as priors for parameters in fundamental models of chemical processes

AbstractModellers of chemical processes with knowledge about plausible parameter values use Bayesian parameter estimation methods to account for their prior beliefs. Some modellers specify prior distributions with finite parameter ranges, such as uniform distributions and truncated normal distributions, because they better account for knowledge about realistic parameter ranges than normal prior distributions with parameter values ranging between and . We derive closed‐form objective functions for Bayesian parameter estimation with truncated normal priors and uniform priors, for the first time, so that parameter estimation can be performed by solving simple optimization problems rather than using complex sampling‐based techniques. A parametric bootstrapping method that considers truncated normal priors and model nonlinearity is proposed to determine 95% confidence intervals and joint confidence regions. A pharmaceutical case study is used to show the effectiveness of the proposed objective functions and bootstrapping methodology. Confidence regions from bootstrapping are similar to linearization‐based confidence regions that do not account for truncation when truncated areas in normal prior distributions are relatively small. More truncation, which corresponds to more‐precise prior knowledge about the parameters, results in smaller joint confidence regions. The proposed methods will be attractive for parameter estimation in complex process models because they can be less computationally intensive than Markov chain Monte Carlo methods that provide similar results.

Model Selection Posterior Predictive Model Checking via Limited‐Information Indices for Bayesian Diagnostic Classification Modeling

AbstractRecently, Bayesian diagnostic classification modeling has been becoming popular in health psychology, education, and sociology. Typically information criteria are used for model selection when researchers want to choose the best model among alternative models. In Bayesian estimation, posterior predictive checking is a flexible Bayesian model evaluation tool, which allows researchers to detect Q‐matrix misspecification. However, model selection methods using posterior predictive checking (PPC) for Bayesian DCM are not well investigated. Thus, this research aims to propose a novel model selection approach using posterior predictive checking with limited‐information statistics for selecting the correct Q‐matrix. A simulation study was conducted to examine the performance of the proposed method. Furthermore, an empirical example was provided to illustrate how it can be used in real scenarios.

Physics-informed tracking of qubit fluctuations

Environmental fluctuations degrade the performance of solid-state qubits but can in principle be mitigated by real-time Hamiltonian estimation down to timescales set by the estimation efficiency. We implement a physics-informed and an adaptive Bayesian estimation strategy and apply them in real time to a semiconductor spin qubit. The physics-informed strategy propagates a probability distribution inside the quantum controller according to the Fokker-Planck equation, appropriate for describing the effects of nuclear spin diffusion in gallium arsenide. Evaluating and narrowing the anticipated distribution by a predetermined qubit probe sequence enables improved dynamical tracking of the uncontrolled magnetic field gradient within the singlet-triplet qubit. The adaptive strategy replaces the probe sequence by a small number of qubit probe cycles, with each probe time conditioned on the previous measurement outcomes, thereby further increasing the estimation efficiency. The combined real-time estimation strategy efficiently tracks low-frequency nuclear spin fluctuations in solid-state qubits, and can be applied to other qubit platforms by tailoring the appropriate update equation to capture their distinct noise sources. Published by the American Physical Society 2024

Inference on exponentiated Rayleigh distribution with constant stress partially accelerated life tests under progressive type-II censoring

This study aims to explore the issues of evaluating the parameters and the accelerating factor based on constant stress for partially accelerating life tests when the potential failure times have an exponentiated Rayleigh distribution. Within the framework of progressive Type-II censoring schemes, we employ the Newton-Raphson algorithm as an iterative methodology to gain the maximum likelihood estimates, accompanied by proof of the existence of these point estimators. We also construct asymptotic confidence intervals for interested parameters and acceleration factors by utilizing the asymptotical characteristics of the maximum likelihood estimators. The Bayesian estimations of unknown parameters are derived by using the independent gamma priors and dependent Gamma-Dirichlet prior on the basis of square error and relatively smooth LINEX loss functions, respectively. Furthermore, we adopt the importance sampling method to compute Bayesian point estimates and the credible intervals with the highest posterior density. To validate the effectiveness of the suggested approaches, a series of simulated experiments are carried out. Lastly, we conduct analyzes on two actual datasets to show the applicability of the suggested techniques.

Ultra-Efficient MCMC for Bayesian Longitudinal Functional Data Analysis

Functional mixed models are widely useful for regression analysis with dependent functional data, including longitudinal functional data with scalar predictors. However, existing algorithms for Bayesian inference with these models only provide either scalable computing or accurate approximations to the posterior distribution, but not both. We introduce a new MCMC sampling strategy for highly efficient and fully Bayesian regression with longitudinal functional data. Using a novel blocking structure paired with an orthogonalized basis reparameterization, our algorithm jointly samples the fixed effects regression functions together with all subject- and replicate-specific random effects functions. Crucially, the joint sampler optimizes sampling efficiency for these key parameters while preserving computational scalability. Perhaps surprisingly, our new MCMC sampling algorithm even surpasses state-of-the-art algorithms for frequentist estimation and variational Bayes approximations for functional mixed models—while also providing accurate posterior uncertainty quantification—and is orders of magnitude faster than existing Gibbs samplers. Simulation studies show improved point estimation and interval coverage in nearly all simulation settings over competing approaches. We apply our method to a large physical activity dataset to study how various demographic and health factors associate with intraday activity. Supplementary materials for this article are available online.

Experimental research on Bayesian methods in large-scale datasets

The current study systematically examines the application of the Bayesian approach or Bayesian estimation methods in large-scale datasets, emphasizing their adaptability as well as predictive prowess across various domains. The study navigates the challenges inherent in the computational efficacy and scalability of these techniques to offer insights into their application in the fields of text analysis, image processing, recommendation systems, and social network analysis. The study uses experimental designs that highlight how well Bayesian methods perform in comparison to more conventional methods, emphasizing how much better they are able to handle uncertainty and incorporate prior knowledge. The future directions and possible enhancements of Bayesian techniques are also discussed, especially with regard to overcoming computational limitations by integrating machine learning and developing sophisticated algorithms. As a crucial tool for modern data analysis and predictive modelling, the study's conclusion upholds the critical role that Bayesian estimation plays in the world of big data.

Fast joint estimation of direction of arrival and towed array shape based on marginal likelihood maximization

This paper addresses the joint estimation problem of directions of arrival of sources and towed array shape. Ocean currents and the maneuvering of the towing platform often cause the towed array to bend rather than stay in a straight line, so the array shape must be estimated to avoid performance degradation due to array mismatch. Existing joint estimation methods suffer from high computational complexity or the need for external sensors installed on the array. Therefore, this paper proposes a fast joint estimation algorithm that solely utilizes received acoustic data. The joint estimation is transformed into a sparse reconstruction optimization problem within the Bayesian estimation framework. Firstly, by decomposing the marginal likelihood function, a closed-form solution for the signal variance in the spatial domain is obtained. Then, the towed array shape is introduced as a hyperparameter into the marginal likelihood function and optimized using the quasi-Newton method. In this way, we accomplish joint estimation and reduce computational complexity. Multi-source simulation results show that the proposed method achieves high resolution and fast computational speed. The robustness and efficiency of the proposed joint estimation method are demonstrated by experimental results in the South China Sea.

Flexible Bayesian estimation of incubation times.

The incubation period is of paramount importance in infectious disease epidemiology as it informs about the transmission potential of a pathogenic organism and helps to plan public health strategies to keep an epidemic outbreak under control. Estimation of the incubation period distribution from reported exposure times and symptom onset times is challenging as the underlying data is coarse. We develop a new Bayesian methodology using Laplacian-P-splines that provides a semi-parametric estimation of the incubation density based on a Langevinized Gibbs sampler. A finite mixture density smoother informs a set of parametric distributions via moment matching and an information criterion arbitrates between competing candidates. Algorithms underlying our method find a natural nest within the EpiLPS package, which has been extended to cover estimation of incubation times. Various simulation scenarios accounting for different levels of data coarseness are considered with encouraging results. Applications to real data on COVID-19, MERS and Mpox reveal results that are in alignment with what has been obtained in recent studies. The proposed flexible approach is an interesting alternative to classic Bayesian parametric methods for estimation of the incubation distribution.

Bayesian estimation of covariate assisted principal regression for brain functional connectivity.

This paper presents a Bayesian reformulation of covariate-assisted principal regression for covariance matrix outcomes to identify low-dimensional components in the covariance associated with covariates. By introducing a geometric approach to the covariance matrices and leveraging Euclidean geometry, we estimate dimension reduction parameters and model covariance heterogeneity based on covariates. This method enables joint estimation and uncertainty quantification of relevant model parameters associated with heteroscedasticity. We demonstrate our approach through simulation studies and apply it to analyze associations between covariates and brain functional connectivity using data from the Human Connectome Project.

Bayesian estimation of gene constraint from an evolutionary model with gene features.

Measures of selective constraint on genes have been used for many applications, including clinical interpretation of rare coding variants, disease gene discovery and studies of genome evolution. However, widely used metrics are severely underpowered at detecting constraints for the shortest ~25% of genes, potentially causing important pathogenic mutations to be overlooked. Here we developed a framework combining a population genetics model with machine learning on gene features to enable accurate inference of an interpretable constraint metric, shet. Our estimates outperform existing metrics for prioritizing genes important for cell essentiality, human disease and other phenotypes, especially for short genes. Our estimates of selective constraint should have wide utility for characterizing genes relevant to human disease. Finally, our inference framework, GeneBayes, provides a flexible platform that can improve the estimation of many gene-level properties, such as rare variant burden or gene expression differences.

Research on methods for estimating reference crop evapotranspiration under incomplete meteorological indicators.

Accurate estimation of reference crop evapotranspiration (ET0) is crucial for farmland hydrology, crop water requirements, and precision irrigation decisions. The Penman-Monteith (PM) model has high accuracy in estimating ET0, but it requires many uncommon meteorological data inputs. Therefore, an ideal method is needed that minimizes the number of input data variables without compromising estimation accuracy. This study aims to analyze the performance of various methods for estimating ET0 in the absence of some meteorological indicators. The Penman-Monteith (PM) model, known for its high accuracy in ET0 estimation, served as the standard value under conditions of adequate meteorological indicators. Comparative analyses were conducted for the Priestley-Taylor (PT), Hargreaves (H-A), McCloud (M-C), and FAO-24 Radiation (F-R) models. The Bayesian estimation method was used to improve the ET estimation model. Results indicate that, compared to the PM model, the F-R model performed best with inadequate meteorological indicators. It demonstrates higher average correlation coefficients (R2) at daily, monthly, and 10-day scales: 0.841, 0.937, and 0.914, respectively. The corresponding root mean square errors (RMSE) are 1.745, 1.329, and 1.423, and mean absolute errors (MAE) are 1.340, 1.159, and 1.196, with Willmott's Index (WI) values of 0.843, 0.862, and 0.859. Following Bayesian correction, R2 values remained unchanged, but significant reductions in RMSE were observed, with average reductions of 15.81%, 29.51%, and 24.66% at daily, monthly, and 10-day scales, respectively. Likewise, MAE decreased significantly, with average reductions of 19.04%, 34.47%, and 28.52%, respectively, and WI showed improvement, with average increases of 5.49%, 8.48%, and 10.78%, respectively. Therefore, the F-R model, enhanced by the Bayesian estimation method, significantly enhances the estimation accuracy of ET0 in the absence of some meteorological indicators.

Different Statistical Inference Algorithms for the New Pareto Distribution Based on Type-II Progressively Censored Competing Risk Data with Applications

In this research, the statistical inference of unknown lifetime parameters is proposed in the presence of independent competing risks using a progressive Type-II censored dataset. The lifetime distribution associated with a failure mode is assumed to follow the new Pareto distribution, with consideration given to two distinct competing failure reasons. Maximum likelihood estimators (MLEs) for the unknown model parameters, as well as reliability and hazard functions, are derived, noting that they are not expressible in closed form. The Newton–Raphson, expectation maximization (EM), and stochastic expectation maximization (SEM) methods are employed to generate maximum likelihood (ML) estimations. Approximate confidence intervals for the unknown parameters, reliability, and hazard rate functions are constructed using the normal approximation of the MLEs and the normal approximation of the log-transformed MLEs. Additionally, the missing information principle is utilized to derive the closed form of the Fisher information matrix, which, in turn, is used with the delta approach to calculate confidence intervals for reliability and hazards. Bayes estimators are derived under both symmetric and asymmetric loss functions, with informative and non-informative priors considered, including independent gamma distributions for informative priors. The Monte Carlo Markov Chain sampling approach is employed to obtain the highest posterior density credible intervals and Bayesian point estimates for unknown parameters and reliability characteristics. A Monte Carlo simulation is conducted to assess the effectiveness of the proposed techniques, with the performances of the Bayes and maximum likelihood estimations examined using average values and mean squared errors as benchmarks. Interval estimations are compared in terms of average lengths and coverage probabilities. Real datasets are considered and examined for each topic to provide illustrative examples.

Estimation of multi-stress strength reliability under progressive first failure censoring using generalized inverted exponential distribution

We investigate the estimation of multi-stress strength reliability under progressively first failure censoring for the generalized inverted exponential distribution using classical and Bayesian approaches. Classical estimation employs maximum likelihood estimators with the Expectation-Maximization algorithm to handle incomplete data. Bayesian estimation utilizes Markov Chain Monte Carlo (MCMC) techniques and Tierney-Kadane's approximation to approximate the posterior distribution. Furthermore, A generalized entropy loss function is employed to measure estimation accuracy. Finally, Validation is provided through Monte Carlo simulations and real-world applications. Findings highlight the robustness and accuracy of Bayesian methods, particularly with informative priors, and underscore the strengths of MCMC techniques and Bayesian Credible Intervals in specific aspects of estimation.

Statistical inference of a series reliability system using shock models with Weibull distribution

AbstractIn this study, we define a series system with non‐independent and non‐identical components using a shock model with sources of fatal shocks. Here, it is assumed that the shocks happen randomly and independently, following a Weibull distribution with various scale and shape parameters. A dependability model with unknown parameters is produced by this process. Making statistical conclusions about the model parameters is the main objective of this research. We apply the maximum likelihood and Bayes approaches to determine the model parameters' point and interval estimates. We shall demonstrate that no analytical solutions to the likelihood equations must be solved to obtain the parameters' maximum likelihood estimates. As a result, we will use the R program to approximate the parameter point and interval estimates. Additionally, we will use the bootstrap‐t and bootstrap‐p methods to approximate the confidence intervals. About the Bayesian approach, we presume that each model parameter is independent and follows a gamma prior distribution with a range of attached hyperparameter values. The model parameters' posterior distribution does not take a practical form. We are unable to derive the Bayes estimates in closed forms as a result. To solve this issue, we use the Gibbs sampler from the Metropolis‐Hasting algorithm based on the Markov chain Monte Carlo method to condense the posterior distribution. To demonstrate the relevance of this research, a real data set application is detailed.

Improving Error Estimates for Evaluating Satellite-Based Atmospheric CO2 Measurement Concepts through Numerical Simulations

To assess the accuracy of satellite monitoring of anthropogenic CO2 emissions, inversions of satellite data in SWIR are usually combined with the assimilation of the total CO2 column into a Kalman filter that reconstructs the sources and sinks of atmospheric CO2. To provide error estimates of the total CO2 column for multi-month assimilation experiments of simulated satellite data, we parametrise these errors using linear regressions. These regression are obtained from a database that links meteorological situations, albedos, and aerosols to the errors in the inversion of the total CO2 column based on simulated satellite data for those conditions. The errors in this database are explicitly computed using the Bayesian estimation formalism, and the linear regressions are optimised by selecting appropriate predictors and predictants. For different levels of measurement noise, error simulations are performed over a period of several months using the albedo and aerosol data from MODIS.

Balancing Performance and Portability: A Study on CsI(Tl) Crystal Sizes for Real-Time Gamma-Ray Spectrum and Dose Monitoring

Current radiation dosimeters sometimes face accuracy limitations or provide only cumulative doses over long periods. To contribute to this area, we developed a portable monitor that measures the energy spectrum and dose of gamma rays in real time. To achieve this, we used an improved sequential Bayesian estimation algorithm. The dose rate was then derived from the energy spectrum by applying a flux-to-dose conversion coefficient. The monitor consists mainly of a CsI(Tl) scintillator and a multi-pixel photon counter (MPPC). In developing this device, we focused on striking a balance between measurement accuracy, ease of use, and portability. As an essential aspect of the research, we investigated the influence of the CsI(Tl) crystal size on the performance of the monitor to determine an optimal size. This was accomplished by calculating the detection efficiency and energy resolution through experimental measurements using standard gamma-ray sources and simulations using MCNP5. Within the scope of the research, detector response functions were created for each crystal size for an energy range of 10 keV to 3 MeV. Considering an optimal balance of detection efficiency and energy resolution alongside a compact size suitable for portable applications, the crystal measuring 2.6 × 2.6 × 1.3 cm3 was deemed preferable.

Residents’ preferences for tourism development in a protected area: a choice experiment

Purpose This study aims to reveal residents’ individual perceptions of nature-based destination development and preferences for infrastructure and tourism superstructure development among communities that rely heavily on wildlife tourism. Design/methodology/approach A discrete choice experiment (DCE) was used among the Maasai community based in the villages and towns near the Maasai Mara National Reserve in Kenya. The attributes included type of tourism accommodation, location of tourism accommodations, types of access roads (tarmac or marram), tourist numbers and desired land-use options (between tourism development, livestock grazing and agriculture). A DCE analysis with hierarchical Bayes estimation was performed. Findings It revealed that the introduction to land-use restrictions and the location of tourism accommodations were the most important attributes for the respondents, with average importance values of 30.36% and 24.02%, respectively. A significant less important attribute was the types of access roads with an average importance of just 8.38%. Cluster analysis revealed widespread heterogeneity in preferences. Research limitations/implications The survey-based DCE was conducted in the Maasai Mara National Reserve, Kenya, and therefore may not be relevant in other contexts. The focus was also only on the residents’ preferences. The findings broaden the knowledge on tourism developments and residents’ support for development and management of protected areas. Practical implications For policymakers, conservation practitioners and tourism businesses, this study provides a source of reference for understanding the development preferences of the Maasai community. In general, the study contributes to a better understanding of local communities in relations to tourism development and residents’ support for developments and management of protected areas (PAs). Originality/value This study fills the gap in the literature on tourism development and residents’ support for developments in PAs by presenting some limits of acceptable and desirable use of PAs among a community that has a complex coexistence with a wildlife tourism destination. It provides an alternative perspective for future research by examining residents’ choice towards destination development and preferences for infrastructure and tourism superstructure development using an experimental approach.

Inference on Weibull inverted exponential distribution under progressive first-failure censoring with constant-stress partially accelerated life test

Accelerated life tests (ALTs) play a pivotal role in life testing experiments as they significantly reduce costs and testing time. Hence, this paper investigates the statistical inference issue for the Weibull inverted exponential distribution (WIED) under the progressive first-failure censoring (PFFC) data with the constant-stress partially ALT (CSPALT) under progressive first-failure censoring (PFFC) data for Weibull inverted exponential distribution (WIED). For classical inference, maximum likelihood (ML) estimates for both the parameters and the acceleration factor are derived. Making use of the Fisher information matrix (FIM), asymptotic confidence intervals (ACIs) are constructed for all parameters. Besides, two parametric bootstrap techniques are implemented. For Bayesian inference based on a proposed technique for eliciting the hyperparameters, the Markov chain Monte Carlo (MCMC) technique is provided to acquire Bayesian estimates. In this context, the Bayesian estimates are obtained under symmetric and asymmetric loss functions, and the corresponding credible intervals (CRIs) are constructed. A simulation study is carried out to assay the performance of the ML, bootstrap, and Bayesian estimates, as well as to compare the performance of the corresponding confidence intervals (CIs). Finally, real-life engineering data is analyzed for illustrative purposes.

Divergences based Bayesian inference with censored data

. In this work, we deal with some Bayesian inference problems in the presence of right censored data. First, we propose a dual ϕ−divergence Bayes type estimators for parametric models and we establish their asymptotic normality. To establish this result, we need a uniform strong law of large numbers that we prove as well. Then, we consider the problem of prior distributions construction for model selection using ϕ−divergences. Finally, we consider the problem of the predictive density estimation on the basis of ϕ−divergences. We apply an expansion result of the generalized Bayesian predictive density on two parametric models widely used in survival analysis, namely the Weibull and the inverse Weibull model, under right censoring. We also check the performances of our proposed methods through simulations and real data applications. The results of these studies show that our proposed dual ϕ−divergence Bayes type estimators are more robust than other Bayesian estimators. Moreover, the generalized Bayesian predictive density performs better than the classical estimative density especially for the inverse Weibull model.