Prior Distribution Of Parameters Research Articles

Comparison of Prior Distributions for the Heterogeneity Parameter in a Rare Events Meta-Analysis of a Few Studies.

Bayesian meta-analysis is a promising approach for rare events meta-analysis. However, the inference of the overall effect in rare events meta-analysis is sensitive to the choice of prior distribution for the heterogeneity parameter. Therefore, it is crucial to assign a convincing prior specification and ensure that it is both plausible and transparent. Various priors for the heterogeneity parameter have been proposed; however, the comparative performance of alternative prior specifications in rare events meta-analysis is poorly understood. Based on a binomial-normal hierarchical model, we conducted a comprehensive simulation study to compare seven heterogeneity prior specifications for binary outcomes, using the odds ratio as the metric. We compared their performance in terms of coverage, median percentage bias, width of the 95% credible interval, and root mean square error (RMSE). We illustrate the results with two recently published rare events meta-analyses of a few studies. The results show that the half-normal prior (with a scale of 0.5), the prior proposed by Turner etal. for the general healthcare setting (without restriction to a specific type of outcome) and for the adverse event setting perform well when the degree of heterogeneity is not relatively high, yielding smaller bias and shorter interval widths with similar coverage and RMSE in most cases compared to other prior specifications. None of the priors performed better when the heterogeneity between-studies were significantly extreme.

Bayesian mixture model for accurate assessment of monthly maximum wind speed: A case study in Gwadar

Assessment of monthly maximum wind speed (MMWS) is essential in various environmental disciplines, including architectural risk assessment, climatology, renewable energy sources, building design, and agricultural activities. The distribution of wind speed is an important module of its development. Presently, several mixing distributions have been suggested for fitting the theoretical maximum wind speed distributions. However, the drawback of various distribution elements is the outcome of numerous fitting presentations, and it is challenging to choose the best form of single distribution to employ when building a hybrid model. In order to solve those difficulties, utilizing the mixtures of Sine-skewed-von Mises distributions (MSS-VM) with various prior parameter distributions is extended in the present study. In this research, the assessment of MMWS employing the maximum-likelihood estimation (MLE) method is taken into consideration for the wind speed data from the Pakistan province of Baluchistan Gwadar station. The results of the Anderson Darling (AD), Chi-square, BIC, and AIC tests, as well as the Kolmogorov Smirnov (KS) test, indicated that MSS-VM was the best distribution for the Gwadar station. The suggested models automatically find the optimum number of elements, in addition to having more appropriate performance. The outcomes demonstrate that MSS-VM performance is more suitable than other heterogeneous and single-distribution models. The design estimations derived using the Bayesian mixture distribution provide a sense of the highest wind speed that will occur across a particular area. It is therefore crucial for designers and policymakers in the planning, designing, and building of various structures.

Penalized Bayesian D-Optimal Designs for Regression Models of Continuous Response

We propose extending the use of desirability functions in Bayesian optimal designs for regression models. This technique generates experimental designs with good statistical inference properties according to Bayesian optimal design theory and practical features, as defined by an investigator. These practical features are defined by a penalty function, using an overall desirability function, which is added to a Bayesian $D$-optimal design criterion to penalize impractical experimental designs. This methodology is illustrated by two examples of regression models: quadratic and exponential. Then, we compare designs obtained for different prior distributions of unknown parameters by efficiency calculations and simulation study. Results show that the $D$-efficiencies of the penalized designs relative to the non-penalized Bayesian D-optimal designs are competitive.

What Do We Know Without the Catalog? Eliciting Prior Beliefs from Experts for Aftershock Models

Abstract Fitting parametric seismological models to earthquake catalogs often comes with numerical challenges, especially when catalogs are small. An alternative way to quantify parameter values for a seismic region is by eliciting expert opinions on the seismological characteristics that each parameter corresponds to. For instance, expert beliefs on aftershock patterns can be formulated into prior distributions for aftershock parameters, for example, for the epidemic-type aftershock sequence (ETAS) model. We illustrate such a method by not only eliciting priors for ETAS parameters for the Pacific Northwest (PNW), a subduction zone with a complex tectonic environment, but also a relatively small catalog. We compare these priors with those suggested by the ETAS literature for global subduction zones, discussing implications for aftershock forecasting for the PNW.

Bayesian merged utilization of GRAPPA and SENSE (BMUGS) for in-plane accelerated reconstruction increases fMRI detection power

In fMRI, capturing brain activity during a task is dependent on how quickly the k-space arrays for each volume image are obtained. Acquiring the full k-space arrays can take a considerable amount of time. Under-sampling k-space reduces the acquisition time, but results in aliased, or “folded,” images after applying the inverse Fourier transform (IFT). GeneRalized Autocalibrating Partial Parallel Acquisition (GRAPPA) and SENSitivity Encoding (SENSE) are parallel imaging techniques that yield reconstructed images from subsampled arrays of k-space. With GRAPPA operating in the spatial frequency domain and SENSE in image space, these techniques have been separate but can be merged to reconstruct the subsampled k-space arrays more accurately. Here, we propose a Bayesian approach to this merged model where prior distributions for the unknown parameters are assessed from a priori k-space arrays. The prior information is utilized to estimate the missing spatial frequency values, unalias the voxel values from the posterior distribution, and reconstruct into full field-of-view images. Our Bayesian technique successfully reconstructed simulated and experimental fMRI time series with no aliasing artifacts while decreasing temporal variation and increasing task detection power.

Can the young water fraction reduce predictive uncertainty in water transit time estimations?

Transit time distributions (TTDs) of streamflow are informative descriptors of catchment hydrological functioning and solute transport mechanisms. Conventional methods for estimating TTDs generally require model calibration against extensive tracer data time series, which are often limited to well-studied experimental catchments. We challenge this limitation and propose an alternative approach that uses the young water fraction (Fywobs), an increasingly used water age metric which represents the proportion of streamflow with a transit time younger than 2-3 months, and that can be robustly estimated with sparsely measured tracer data. To this end, we conducted a proof of concept study by modelling TTDs using StorAge Selection (SAS) functions with oxygen isotopes (δ18O) measurements for 23 diverse catchments in Germany. In a Monte-Carlo approach, we computed the (averaged) marginal TTDs of a prior parameter distribution and derived a model-based Fyw (Fywsim). We compared Fywsim with Fywobs, obtained from δ18O measurements, and constrained the prior SAS parameters distribution. Subsequently, we derived a posterior distribution of parameters and resulting model simulations. Our findings showed that using Fywobs to constrain the model effectively reduced parameter equifinality and simulation uncertainty. However, the value of Fywobs on reducing model uncertainty varied across sites, with larger values (Fywobs≥0.10) leading to simulations with a narrower uncertainty band and higher model efficiency, whilst smaller values (Fywobs≤0.05) had limited influence on reducing model output uncertainty. We discussed the potential and limitations of combining SAS functions with Fywobs, and considered broader implications of this approach for enhancing our understanding of catchment functioning and water quality status.

Damage inference of statically deterministic structural trusses is carried out by Bayesian updating

In the face of harsh environments, aging construction materials, and increased traffic, the performance of Bridges in service continues to deteriorate. To avoid the serious consequences caused by structural damage, it is of great significance to infer the damage of the truss structure and evaluate the reliability of the bridge. In this paper, a random sampling process is used to characterize the resistance degradation, and the historical load test information and Bayesian method are used to update the resistance of serving Bridges. The modeling of structural mechanics solver determines the maximum axial force on the damaged structural member by loading different nodes. Secondly, node displacement is calculated. Then, the displacement results are taken as new information as the parameters of prior distribution for Bayesian update, and finally, the maximum probability of structural damage is obtained. The results show that the probability of EA values between 1.34 and 1.38 is the greatest. The uncertainty and variability in the process of resistance degradation of reinforced concrete bridge members can be described more reasonably by the Bayesian updating process. By integrating the real resistance degradation information detected during bridge service into practical application, and using the research method in this paper to analyze and calculate, Bridges' future service status and remaining service life can be predicted more accurately.

Assessing the Role of Patient Generation Techniques in Virtual Clinical Trial Outcomes.

Virtual clinical trials (VCTs) are growing in popularity as a tool for quantitatively predicting heterogeneous treatment responses across a population. In the context of a VCT, a plausible patient is an instance of a mathematical model with parameter (or attribute) values chosen to reflect features of the disease and response to treatment for that particular patient. A number of techniques have been introduced to determine the set of model parametrizations to include in a virtual patient cohort. These methodologies generally start with a prior distribution for each model parameter and utilize some criteria to determine whether a parameter set sampled from the priors should be included or excluded from the plausible population. No standard technique exists, however, for generating these prior distributions and choosing the inclusion/exclusion criteria. In this work, we rigorously quantify the impact that VCT design choices have on VCT predictions. Rather than use real data and a complex mathematical model, a spatial model of radiotherapy is used to generate simulated patient data and the mathematical model used to describe the patient data is a two-parameter ordinary differential equations model. This controlled setup allows us to isolate the impact of both the prior distribution and the inclusion/exclusion criteria on both the heterogeneity of plausible populations and on predicted treatment response. We find that the prior distribution, rather than the inclusion/exclusion criteria, has a larger impact on the heterogeneity of the plausible population. Yet, the percent of treatment responders in the plausible population was more sensitive to the inclusion/exclusion criteria utilized. This foundational understanding of the role of virtual clinical trial design should help inform the development of future VCTs that use more complex models and real data.

A variational Bayesian based robust filter for unknown measurement bias and inaccurate noise statistics

In many practical fields, the unknown time-varying measurement biases (additive and multiplicative bias) and heavy-tailed measurement noise caused by some unpredictable anomalous behaviors may degrade the performance of conventional Kalman filter seriously. To solve the state estimation problem of systems with time-varying measurement biases and heavy-tailed measurement noise, this paper proposes a new variational Bayesian (VB) based robust filter. Firstly, the non-Gaussian measurement likelihood probability density function (ML-PDF) with multiplicative and additive measurement bias is built. Then, the conjugate prior distributions for unknown bias and noise scale parameters are selected, and the VB method is utilized to jointly infer the system state, unknown measurement biases and inaccurate measurement noise covariance matrix. Finally, a VB based robust filter is derived and its effectiveness is verified by the numerical simulations.

Bayesian Method for Quality Control with Weibull Distribution

Weibull distribution is one of the continous probability density function. This distribution is known as a flexible distribution. One of it’s flexibility is can be transformed into other distributions such as exponential distribution depends on parameter selected. Similar to other distribution, Weibull distribution also characterized by cumulative distribution function, mean, variance and moment generating function. One of the well-known application of this distribution is in the field of quality control utilizing reliability data and the well-known tool in quality control is control chart. Therefore, reliability data is not follow normal distribution then Shewhart control chart cannot be applied. To solve this problem, control chart can be formed with the control limit is obtained by Bayesian method. In applying Bayesian method, prior distribution of shape parameter is assumed to be uniform distribution and variables for reliability is assumed to be invers of Weibull distribution. The prior distribution is combine with likelihood function then posterior distribution is obtained which in then used as the control limits by find it’s mean.

Basin stability for updating system uncertainties.

In this paper, we propose an application of the basin stability tool which allows us to update the information on the system properties under parameter uncertainties. The concept is presented using a classical mechanical setup of coupled pendula, exchanging the energy via the supporting structure. Depending on the support parameters, the model can exhibit different types of coexisting synchronous patterns as well as remaining desynchronized. We calculate basin stability maps of particular behaviors and combine them with prior parameter distributions using Bayesian inference. The obtained posterior distributions, based on the attractor occurrence, update our knowledge on the system properties in the terms of probabilities. We also underline the problem of evaluating basin stability close to the existence borders, comparing the classical approach of fixed parameters with the one involving variations. The differences between the estimation methods can have a crucial meaning for the discussed application and should be considered carefully. The results presented in this paper uncover ways of applying the basin stability concept, which can be used to study the properties of complex dynamical systems from a probability perspective.

A multiobjective optimization framework for site investigation program based on Bayesian approach and NSGA‐II

AbstractSite investigation provides essential geotechnical parameter information for analysis and design. However, three conflicting objectives, namely exploration effort, robustness and parameter uncertainty, pose a challenge to the development of an optimal site investigation program. In this study, a three objective optimization framework for the site investigation program is proposed based on the Bayesian approach and the non‐dominated sorting genetic algorithm (NSGA‐II). The only inputs required by the proposed framework are prior distribution of geotechnical parameters and error information. The prior distribution of geotechnical parameters is derived from integrating engineering experience and measurements from basic exploration boreholes. The error information is obtained based on literature and expert judgment related to the specific project. Firstly, a design pool of candidate investigation programs is generated using Bayesian approach to determine the locations and number of exploration boreholes. The NSGA‐II is then applied to identify the optimal program that balances lower cost, higher robustness, and lower uncertainty. The proposed multiobjective optimization framework is illustrated and validated through a real site investigation case in Chongqing, China, aimed at determining the ultimate bearing capacity of the rock foundation. The spatial correlation of parameters within the study area is also considered. The optimal program is represented by the location and number of exploration boreholes. By comparing measurements with predictions from different site investigation programs, the efficiency of the proposed multiobjective framework is demonstrated. Additionally, the influence of engineering experience and random field modeling on the investigation program is discussed.

Inverse heat transfer problem for the characterization of nanofluids produced with different types of palladium nanoparticles

The objective of this work is the measurement of physical properties of distilled water nanofluids containing palladium nanocubes, palladium cerium oxide nanoparticles, and their respective hydrides. Due to their biocompatibility and favorable photothermal effects, palladium nanoparticles can be used to promote localized absorption of external energy sources in the thermal treatment of cancer, aiming at a thermal damage constrained to the tumor region without significant effects to the healthy tissues. An inverse problem is solved here within the Bayesian framework of statistics with the Markov Chain Monte Carlo method, by using nonintrusive transient measurements taken with an infrared camera during the heating of different water-based nanofluids. The mathematical model used in this work takes into account natural convection effects, due to the nonuniform heat source caused by the diode-laser that heats the nanofluids. Prior distributions for the model parameters were selected based on additional independent measurements, theoretical models and by the careful implementation of the experiments. The proposed model and the estimated parameters were validated, with an excellent agreement between the measured temperatures and those obtained from stochastic simulations during the solution of the inverse problem.

Deep network fault diagnosis for imbalanced small-sized samples via a coupled adversarial autoencoder based on the Bayesian method.

Deep network fault diagnosis methods heavily rely on abundant labeled data for effective model training. However, small-sized samples and imbalanced samples often lead to insufficient features, resulting in accuracy degradation and even instability in the diagnosis model. To address this challenge, this paper introduces a coupled adversarial autoencoder (CoAAE) based on the Bayesian method. This model aims to solve the issue of insufficient samples by generating fake samples and integrating them with the original ones. Within the CoAAE framework, the probability density distribution of the original data is captured using an encoder and fake samples are generated by random sampling from this distribution and decoding them. This process is the adversarial interaction between the encoder and a classifier to obtain the prior distribution of the encoder's parameters. The encoder's parameters are updated through the decoder's reconstruction process, leading to the posterior distribution. Concurrently, the decoder is trained to enhance its ability to reconstruct samples accurately. To address the imbalance in the original samples, a parallel coupled network is employed. This network shares the weights of the extraction layer in the encoder, enabling it to learn the joint distribution between fault-related and normal samples. To evaluate the effectiveness of the proposed data augmentation method, experiments were conducted on a bearing database from Case Western Reserve University using ResNet18 as the deep learning diagnosis model representative. The results demonstrate that CoAAE can effectively augment imbalanced datasets and outperform other advanced methods.

Normalizing flow based uncertainty estimation for deep regression analysis

Uncertainty estimation is a critical component of building safe and reliable machine learning models. Accurate estimation of uncertainties is essential for identifying and mitigating potential risks and ensuring that machine learning systems operate reliably in real-world scenarios. Various approaches, such as ensemble and Bayesian neural networks have been developed by sampling probability predictions from submodels, which is computationally expensive. At present, these techniques are incapable of precisely delineating the boundary separating in-distribution (ID) and out-of-distribution (OOD) data. To fill up this research gap, this paper presents a normalizing flow based framework to directly predict parameters of prior distributions over the probability with a neural network, the proposed model is able to effectively differentiate between ID and OOD data in regression problems. The posterior distributions learned by the model precisely represent uncertainties for OOD data based solely on ID data, without the need for OOD data during training. This approach has shown promising results in a number of applications, including image depth estimation and image adversarial attacks.

Locally, Bayesian and non parametric Bayesian optimal designs for unit exponential regression model

. This study introduces optimal designs for the unit exponential (UE) non linear model using local, Bayesian, and non parametric Bayesian approaches. In the local approach, optimal designs were derived by substituting initial estimates for the unknown parameters. However, recognizing the inefficiency of these designs when initial estimates are distant from their true values, we adopted a Bayesian approach, employing the D-optimal criterion with uniform and truncated normal prior distributions for unknown parameters. In situations lacking informative or historical knowledge of parameters, a non parametric Bayesian approach was employed, incorporating the Dirichlet Process (DP) prior to the space of distribution functions. Finally, the efficiency of various optimal designs was analyzed for comparison.

Bayesian multinomial model with inference based on Pólya-Gamma augmentation

Statistical inferences from multi-categorical data are often encountered in many scientific fields. It is customary in the literature to model the probability of occurrence of a certain category on some risk factors. However, this work presents a framework to model the stick-breaking probabilities of a multi-categorical event on risk factors. In addition, we adopted a Pólya-Gamma data augmentation to arrive at a conjugate prior distribution for the model parameters and developed a cheap Gibbs sampling algorithm for model estimation. The proposed model allows for flexible and computationally efficient modeling of the conditional properties of an ordered categorical process and the incorporation of dependencies between the different categories of the outcome. The framework was used to model data derived from the Nigeria Demographic and Health Survey to study the spatial patterns and risk factors of maternal malnutrition in Nigeria. The findings identified the determinant factors and showed heterogeneity in the spatial patterns of maternal malnutrition prevalence across Nigeria. Findings from the analysis could be relevant for designing intervention programs and guiding policy-making, such as in the allocation of limited resources across the country.

Sensitivity analysis of prior distributions in Bayesian graphical modeling: Guiding informed prior choices for conditional independence testing

Bayesian analysis methods provide a significant advancement in network psychometrics, allowing researchers to use the edge inclusion Bayes factor for testing conditional independence between pairs of variables in the network. Using this methodology requires setting prior distributions on the network parameters and on the network’s structure. However, the impact of both prior distributions on the inclusion Bayes factor is underexplored. In this paper, we focus on a specific class of Markov Random Field models for ordinal and binary data. We first discuss the different choices of prior distributions for the network parameters and the network structure, and then perform an extensive simulation study to assess the sensitivity of the inclusion Bayes factor to these distributions. We pay particular attention to the effect of the scale of the prior on the inclusion Bayes factor. To improve the accessibility of the results, we also provide an interactive Shiny app. Finally, we present the R package simBgms, which provides researchers with a user-friendly tool to perform their own simulation studies for Bayesian Markov Random Field models. All of this should help researchers make more informed, evidence-based decisions when preparing to analyze empirical data using network psychometric models.

Bayesian random-effects meta-analysis with empirical heterogeneity priors for application in health technology assessment with very few studies.

In Bayesian random-effects meta-analysis, the use of weakly informative prior distributions is of particular benefit in cases where only a few studies are included, a situation often encountered in health technology assessment (HTA). Suggestions for empirical prior distributions are available in the literature but it is unknown whether these are adequate in the context of HTA. Therefore, a database of all relevant meta-analyses conducted by the Institute for Quality and Efficiency in Health Care (IQWiG, Germany) was constructed to derive empirical prior distributions for the heterogeneity parameter suitable for HTA. Previously, an extension to the normal-normal hierarchical model had been suggested for this purpose. For different effect measures, this extended model was applied on the database to conservatively derive a prior distribution for the heterogeneity parameter. Comparison of a Bayesian approach using the derived priors with IQWiG's current standard approach for evidence synthesis shows favorable properties. Therefore, these prior distributions are recommended for future meta-analyses in HTA settings and could be embedded into the IQWiG evidence synthesis approach in the case of very few studies.

Resolution-independent generative models based on operator learning for physics-constrained Bayesian inverse problems

The Bayesian inference approach is widely used to solve inverse problems due to its versatile and natural ability to handle ill-posedness. However, there are often challenges when dealing with situations involving continuous fields or parameters with large-resolution discrete representations. Furthermore, the prior distribution of the unknown parameters is also commonly difficult to determine. Therefore, in this study, an operator learning-based generative adversarial network (OL-GAN) is proposed and integrated into the Bayesian inference framework to address these issues. Compared with classical Bayesian approaches, the distinctive characteristic of the proposed method is that it learns the joint distributions of parameters and responses. By using the trained generative model to handle the prior in Bayes’ rule, the posteriors of the unknown parameters can theoretically be approximated by any sampling algorithms (e.g., Markov chain Monte Carlo, MCMC) under the proposed framework. More importantly, efficient sampling can be implemented in a low-dimensional latent space shared by the components of the joint distribution. The latent space is typically a simple and easy-to-sample distribution (e.g., Gaussian, uniform), which significantly reduces the computational cost associated with the Bayesian inference while avoiding prior selection. Furthermore, the generator is resolution-independent due to the incorporation of operator learning. Predictions can thus be obtained at desired coordinates, and inversions can be performed even if the observation data are misaligned with the training data. Finally, the effectiveness of the proposed method is validated through several numerical experiments.