Prior Distribution Of Parameters Research Articles

Non-probabilistic credible reliability analysis of the composite laminate

For the reliability analysis of composite laminate structures, there is a problem of difficult data acquisition. Traditional reliability analysis methods require a large amount of data to maintain calculation accuracy. Therefore, the traditional reliability analysis method is not applicable to composite laminates. For the above-mentioned problems, this paper presents a method to calculate the credible reliability of composite laminate structures using the Bayesian theorem. The method is distinguished from the traditional uncertainty analysis methods. The likelihood function and the newly introduced samples are first used to correct the prior distribution of the uncertain parameters. Furthermore, the posterior distribution, containing information from both the sample and prior distributions, can be obtained. Ultimately, based on the acquired posterior distribution, the probability density function can be updated with newly introduced samples. Using the Hashin failure criterion, the failure probabilities of the fibers and matrix of the composite laminate structure are quantified, respectively. Then, the credible reliability of the composite laminate is obtained. Numerical example results show the proposed method can maintain high accuracy with fewer samples and demonstrate the validity of the model. For the numerical calculations, the deterministic parameters of the distribution function and the functional distribution parameters were verified, composite laminates and aircraft wing structures were analyzed. The experimental results show that the error between the probability density function of the parameters and the probability density function predicted by the sample is less than 0.045. Compared with the traditional reliability analysis methods for composite structures, this method can effectively measure the credible reliability of composite laminate structures with fewer samples and similar accuracy. The method enables real-time updating of the parameter probability density function by newly introduced samples and improves the confidence level.

A dynamic power prior approach to non-inferiority trials for normal means.

Non-inferiority trials compare new experimental therapies to standard ones (active control). In these experiments, historical information on the control treatment is often available. This makes Bayesian methodology appealing since it allows a natural way to exploit information from past studies. In the present paper, we suggest the use of previous data for constructing the prior distribution of the control effect parameter. Specifically, we consider a dynamic power prior that possibly allows to discount the level of borrowing in the presence of heterogeneity between past and current control data. The discount parameter of the prior is based on the Hellinger distance between the posterior distributions of the control parameter based, respectively, on historical and current data. We develop the methodology for comparing normal means and we handle the unknown variance assumption using MCMC. We also provide a simulation study to analyze the proposed test in terms of frequentist size and power, as it is usually requested by regulatory agencies. Finally, we investigate comparisons with some existing methods and we illustrate an application to a real case study.

Optimizing prior distribution parameters for probabilistic prediction of remaining useful life using deep learning

In this study, a deep learning-based probabilistic remaining useful life (RUL) prediction model is proposed to improve the strong prior limitations of traditional probabilistic RUL prediction methods through a flexible prior distribution and strategy for sequential optimization of hyperparameters with regularization factor. It enables output richer probabilistic lifetime density distributions and confidence intervals with various parameters and overcome the problem of poor accuracy of short RUL predictions to some extent. Eventually, the model is effectively validated on a benchmark dataset, and the experimental results show that the probabilistic lifetime prediction model with optimized prior distribution parameters significantly improves prediction performance and demonstrates good learning performance and robustness of test results compared with traditional point estimation methods and parameter-free models. This study informs maintenance decisions and reliability assessments in engineering systems and guides the research and application of probabilistic-based prediction methods in deep learning framework.

A sample size-dependent prior strategy for bridging the Bayesian-frequentist gap in point null hypothesis testing

Bayes factor, as a measure of the evidence provided by the data in favor of one hypothesis against its alternative, can be highly sensitive to the prior distributions of parameters involved in the hypotheses as well as to the sample size. This may cause a noticeable difference between the Bayesian and classical (frequentist) hypothesis testing results. In the worst-case scenario, the two results are in conflict, which is termed the Jeffreys-Lindley paradox. In this article, we propose a sample size-dependent prior strategy to bridge the Bayesian-frequentist gap from a decision-theoretical perspective. The central idea behind the proposed strategy is to adaptively adjust prior distributions for the parameters in line with the sample size to manage the risk of type I error in Bayesian hypothesis testing at the same level as that prespecified in frequentist hypothesis testing. The proposed strategy is inspired by the work of Maurice Stevenson Bartlett (M.S. Bartlett, A comment on D. V. Lindley’s statistical paradox, Biometrika, 44, 533–534, 1957), who suggested a sample size-dependent prior to make the Bayes factor independent of the sample size. In contrast to his work, we propose a strategy that leverages the use of sample size-dependent priors in Bayesian hypothesis testing and risk management when deciding the two hypotheses. To demonstrate the effectiveness of the proposed strategy, normal mean tests in the cases that (i) the variance is known (z-test) and (ii) the variance is unknown (t-test) are examined. It turns out that the Bayesian testing results coming out from the proposed strategy become consistent with their frequentist counterparts and the Jeffreys-Lindley paradox disappears.

Comparison between pystan and numpyro in Bayesian item response theory: evaluation of agreement of estimated latent parameters and sampling performance.

The purpose of this study is to compare two libraries dedicated to the Markov chain Monte Carlo method: pystan and numpyro. In the comparison, we mainly focused on the agreement of estimated latent parameters and the performance of sampling using the Markov chain Monte Carlo method in Bayesian item response theory (IRT). Bayesian 1PL-IRT and 2PL-IRT were implemented with pystan and numpyro. Then, the Bayesian 1PL-IRT and 2PL-IRT were applied to two types of medical data obtained from a published article. The same prior distributions of latent parameters were used in both pystan and numpyro. Estimation results of latent parameters of 1PL-IRT and 2PL-IRT were compared between pystan and numpyro. Additionally, the computational cost of the Markov chain Monte Carlo method was compared between the two libraries. To evaluate the computational cost of IRT models, simulation data were generated from the medical data and numpyro. For all the combinations of IRT types (1PL-IRT or 2PL-IRT) and medical data types, the mean and standard deviation of the estimated latent parameters were in good agreement between pystan and numpyro. In most cases, the sampling time using the Markov chain Monte Carlo method was shorter in numpyro than that in pystan. When the large-sized simulation data were used, numpyro with a graphics processing unit was useful for reducing the sampling time. Numpyro and pystan were useful for applying the Bayesian 1PL-IRT and 2PL-IRT. Our results show that the two libraries yielded similar estimation result and that regarding to sampling time, the fastest libraries differed based on the dataset size.

An informative prior distribution on functions with application to functional regression

We provide a prior distribution for a functional parameter so that its trajectories are smooth and vanish on a given subset. This distribution can be interpreted as the distribution of an initial Gaussian process conditioned to be zero on a given subset. Precisely, we show that the initial Gaussian process is the sum of the conditioned process and an independent process with probability one and that all the processes have the same almost sure regularity. This prior distribution is use to provide an interpretable estimate of the coefficient function in the linear scalar‐on‐function regression; by interpretable, we mean a smooth function that may possibly be zero on some intervals. We apply our model in a simulation and real case studies with two different priors for the null region of the coefficient function. In one case, the null region is known to be an unknown single interval. In the other case, it can be any unknown unions of intervals.

Bayesian ridge estimators based on copula-based joint prior distributions for logistic regression parameters

Ridge regression was originally proposed as an alternative to ordinary least-squares regression to address multicollinearity in linear regression and was later extended to logistic and Cox regressions. The ridge estimator is interpreted as the Bayesian posterior mean or median in the Bayesian framework when the regression coefficients have multivariate normal priors. We previously proposed using vine copula-based joint priors on regression coefficients in linear and Cox regressions, including an interaction that promotes the use of ridge regression because the interaction term can result in multicollinearity. We showed that vine copula-based priors improve the estimation accuracy over the multivariate normal prior, and they would be a promising approach in other regression types, such as logistic regression. In this study, we focus on a case involving two covariates and their interaction terms, and propose a vine copula-based prior for Bayesian ridge estimators under a logistic model. Simulation and data analysis results show that Archimedean (Clayton and Gumbel) copula priors are superior to other priors (the Gaussian copula and trivariate normal priors) in the presence of multicollinearity.

The influence of inter-regional delays in generating large-scale brain networks of phase synchronization

Large-scale networks of phase synchronization are considered to regulate the communication between brain regions fundamental to cognitive function, but the mapping to their structural substrates, i.e., the structure-function relationship, remains poorly understood. Biophysical Network Models (BNMs) have demonstrated the influences of local oscillatory activity and inter-regional anatomical connections in generating alpha-band (8–12 Hz) networks of phase synchronization observed with Electroencephalography (EEG) and Magnetoencephalography (MEG). Yet, the influence of inter-regional conduction delays remains unknown. In this study, we compared a BNM with standard “distance-dependent delays”, which assumes constant conduction velocity, to BNMs with delays specified by two alternative methods accounting for spatially varying conduction velocities, “isochronous delays” and “mixed delays”. We followed the Approximate Bayesian Computation (ABC) workflow, i) specifying neurophysiologically informed prior distributions of BNM parameters, ii) verifying the suitability of the prior distributions with Prior Predictive Checks, iii) fitting each of the three BNMs to alpha-band MEG resting-state data (N = 75) with Bayesian optimization for Likelihood-Free Inference (BOLFI), and iv) choosing between the fitted BNMs with ABC model comparison on a separate MEG dataset (N = 30). Prior Predictive Checks revealed the range of dynamics generated by each of the BNMs to encompass those seen in the MEG data, suggesting the suitability of the prior distributions. Fitting the models to MEG data yielded reliable posterior distributions of the parameters of each of the BNMs. Finally, model comparison revealed the BNM with “distance-dependent delays”, as the most probable to describe the generation of alpha-band networks of phase synchronization seen in MEG. These findings suggest that distance-dependent delays might contribute to the neocortical architecture of human alpha-band networks of phase synchronization. Hence, our study illuminates the role of inter-regional delays in generating the large-scale networks of phase synchronization that might subserve the communication between regions vital to cognition.

Interface State Density Prediction between an Insulator and a Semiconductor by Gaussian Process Regression Models for a Modified Process.

During data-driven process condition optimization on a laboratory scale, only a small-size data set is accessible and should be effectively utilized. On the other hand, during process development, new operations are frequently inserted or current operations are modified. These accessible data sets are somewhat related but not exactly the same type. In this study, we focus on the prediction of the quality of the interface between an insulator and GaN as a semiconductor for the potential application of GaN power semiconductor devices. The quality of the interface was represented as the interface state density, Dit, and the inserted operation to the process was the ultraviolet (UV)/O3-gas treatment. Our retrospective evaluation of model-building approaches for Dit prediction from a process condition revealed that for the UV/O3-treated interfaces, data of interfaces without the treatment contributed to performance improvement. Such performance improvement was not observed when using a data set of Si as the semiconductor. As a modeling method, the automatic relevance vector-based Gaussian process regression with the prior distribution of the length-scale parameters exhibited a relatively high predictive performance and represented a reasonable uncertainty of prediction as reflected by the distance to the training data set. This feature is a prerequisite for a potential application of Bayesian optimization. Furthermore, hyperparameters in the prior distribution of the length-scales could be optimized by leave-one-out cross-validation.

Discussion of specifying prior distributions in reliability applications

AbstractThe paper Specifying Prior Distributions in Reliability Applications mainly provides an overview of methods for selecting non‐informative prior distributions for parameters of basic lifetime distributions, as often used in reliability analyses. This discussion raises some related issues and comments on opportunities beyond basic Bayesian statistical methods which may be useful in reliability scenarios. The main emphasis in this discussion is on practical reliability analyses with few data available, where there is often need for informative priors rather than for non‐informative priors, in order to take expert judgement into account. Furthermore, while rather abstract considerations of non‐informativeness of prior distributions is of theoretic interest, in most practical scenarios one aims at decision support, and the influence of assumed priors on the final decisions should be considered, ideally with robustness of the final decision with regard to all priors which are deemed to be reasonable.

The influence of different back analysis conditions on the stochastic back analysis of arch dam material Parameters: A case study

It is very important to back analyse arch dam material parameters through monitoring data. The selection of the monitoring points, prior distribution of the material parameters and likelihood function have a significant impact on the stochastic back analysis of arch dam material parameters based on Bayesian inference. In this paper, the sensitivity analysis method is used, and the sensitivity of the comprehensive elastic modulus of dam concrete and bedrock for arch dam deformation is analysed. Through high-sensitivity monitoring points and all monitoring points in the arch dam, the influence of the monitoring point selection on the stochastic back analysis of the arch dam material parameters is studied. Considering the prior distribution of the material parameters and likelihood function forms, the influence of different prior distributions and likelihood function forms on the stochastic back analysis of the arch dam material parameters is studied based on high sensitivity measuring points. The example have shown that the sensitivity analysis can be used for optimizing the selection of monitoring points. Different parameter prior distributions and likelihood function forms have certain influences on the stochastic back analysis of the arch dam material parameters. This paper provides a reference for selecting the stochastic inversion conditions for the arch dam material parameters.

A Bayesian Framework for Estimating Weibull Distribution Parameters: Applications in Finance, Insurance, and Natural Disaster Analysis

This research presents a Bayesian framework for parameter estimation in the two-parameter Weibull distribution, with applications in finance and investment data analysis. The Weibull distribution is widely used for modeling stock pricing movements and making uncertain predictions in financial datasets. The proposed Bayesian approach assumes a gamma prior distribution for the scale parameter, with a known shape parameter. A simulation study using simulated financial data compares the Bayesian method with maximum likelihood estimators in terms of accuracy, error accumulation, and computational time across various sample sizes and parameter values. Results indicate the Bayesian approach performs similarly to maximum likelihood for small samples, while demonstrating computational efficiency for larger financial datasets. The proposed Bayesian model's application to simulated financial data showcases its practical relevance in real-world scenarios. This Bayesian framework offers a valuable tool for handling uncertainty and making informed decisions in financial data analysis, providing robust parameter estimation and uncertainty quantification in finance and investment domains.

Bayesian optimal sensor placement for parameter estimation under modeling and input uncertainties

A Bayesian optimal sensor placement (OSP) framework for parameter estimation in nonlinear structural dynamics models is proposed, based on maximizing a utility function built from appropriate measures of information contained in the input–output response time history data. The information gain is quantified using Kullback–Leibler divergence (KL-div) between the prior and posterior distribution of the model parameters. The design variables may include the type and location of sensors. Asymptotic approximations, valid for large number of data, provide valuable insight into the measure of information. Robustness to uncertainties in nuisance (non-updatable) parameters associated with modeling and excitation uncertainties is considered by maximizing the expected information gain over all possible values of the nuisance parameters. In particular, the framework handles the case where the excitation time history is measured by installed sensors but remains unknown at the experimental design phase. Introducing stochastic excitation models, the expected information gain is taken over the large number of uncertain parameters used to model the random variability in the input time histories. Monte Carlo or sparse grid methods estimate the multidimensional probability integrals arising in the formulation. Heuristic algorithms are used to solve the optimization problem. The effectiveness of the method is demonstrated for a multi-degree of freedom (DOF) spring–mass chain system with restoring elements that exhibit hysteretic nonlinearities.

A federated learning differential privacy algorithm for non-Gaussian heterogeneous data

Multi-center heterogeneous data are a hot topic in federated learning. The data of clients and centers do not follow a normal distribution, posing significant challenges to learning. Based on the assumption that the client data have a multivariate skewed normal distribution, we improve the DP-Fed-mv-PPCA model. We use a Bayesian framework to construct prior distributions of local parameters and use expectation maximization and pseudo-Newton algorithms to obtain robust parameter estimates. Then, the clipping algorithm and differential privacy algorithm are used to solve the problem in which the model parameters do not have a display solution and achieve privacy guarantee. Furthermore, we verified the effectiveness of our model using synthetic and actual data from the Internet of vehicles.

Applying Monte Carlo Simulations to a Small Data Analysis of a Case of Economic Growth in COVID-19 Times.

Studies on the going-on COVID-19 pandemic face small sample issues. In this context, Bayesian estimation is considered a viable alternative to frequentist estimation. Demonstrating the Bayesian approach's advantage in dealing with this problem, our research conducted a case study concerning ASEAN economic growth during the COVID-19 pandemic. By using Monte Carlo standard errors and interval hypothesis testing to check parameter bias within a Bayesian MCMC simulation study, the author obtained significant conclusions as follows: first, in insufficient sample sizes, in contrast to frequentist estimation, the Bayesian framework can offer meaningful results, that is, expansionary monetary and contractionary fiscal policies are positively associated with economic growth; second, in the face of a small sample, by incorporating more information into prior distributions for the model parameters, Bayesian Monte Carlo simulations perform so far better than naïve Bayesian and frequentist estimation; third, in case of a correctly specified prior, the inferences are robust to different prior specifications. The author strongly recommends applying specific informative priors to Bayesian analyses, particularly in small sample investigations.

Assessment of the Bering-Chukchi-Beaufort Seas stock of bowhead whales using Bayesian model averaging

Bayesian estimation methods are used to fit an age- and sex-structured population model to available data on abundance and stageproportions (i.e. calves/mature animals in the population) for the Bering-Chukchi-Beaufort Seas stock of bowhead whales (Balaena mysticetus). The analyses consider three alternative population modelling approaches: (1) modelling the entire population trajectory from 1848, using the ‘backwards’ method where the trajectory is back-calculated based on assigning a prior distribution to recent abundance; (2) modelling only the recent population trajectory, using the ‘forwards from recent abundance’ method, where the population is projected forwards from a recent year and the abundance in that year is not assumed to be at carrying capacity; and (3) a version of (2) that ignores density-dependence. The ‘backwards’ method leads to more precise estimates of depletion level. In contrast, the ‘forwards from recent abundance’ method provides an alternative way of calculating catch-related quantities without having to assume that the catch record is known exactly from 1848 to the present, or having to assume that carrying capacity has not changed since 1848. Not only are all three models able to fit the abundance data well, but each is also able to remain consistent with available estimates of adult survival and age of sexual maturity. Sensitivity to the stage-proportion data and the prior distributions for the life history parameters indicates that use of the 1985 stage-proportion data has the greatest effect on the results, and that those data are less consistent with data on trends in abundance and age of sexual maturity. The analyses indicate that the population has approximately doubled in size since 1978, and the ‘backwards’ analyses suggest that the population may be approaching carrying capacity, although there is no obvious sign in the data that the population growth rate has slowed. Bayes factors are calculated to compare model fits to the data. However, there is no evidence for selecting one model over another, and furthermore, the models considered in this study result in different posterior distributions for quantities of interest to management. Posterior model probabilities are therefore calculated and used as weights to construct Bayesian model-averaged posterior distributions for outputs shared among models to take this ambiguity into account. This study represents the first attempt to explicitly quantify model uncertainty when conducting a stock assessment of bowhead whales.

Frequency-dependent site amplification functions for key geological units in Iceland from a Bayesian hierarchical model for earthquake strong-motions

A fundamental modeling assumption in engineering seismology is that seismic ground motion is affected by source, propagation path and site effects. Contrary to the first two, comprehensive studies on site effects, the localized amplification effects on seismic ground motion due to the uppermost geological structure, have been limited in Iceland. As a result, site amplification has been modeled in ground motion models in a simplistic and qualitative way. In this study, we address this limitation by using the new Bayesian hybrid GMM proposed by Kowsari et al. (2020) to model the recorded ground motions at regional strong-motion stations in southwest Iceland, specifically the peak ground acceleration and pseudo-acceleration response spectra as a function of oscillator period (i.e., PGA and PSA at T=0.01-3.0 s), earthquake magnitude, distance and hypocentral depth, and at first, we omit a site-term. Then, we incorporate the GMM in a Bayesian Hierarchical Model (BHM) that allows prior distributions of model parameters to be used and we partition the model residuals into various terms associated with the source, propagation path and station terms. Through Bayesian inference of the model parameters the results show that the station term residuals reveal a frequency-dependent site amplification at the recording sites. By inspecting this systematic behaviour, we propose four groups of station terms by their characteristics. We then carefully inspect the station locations in terms of their associated geological unit and observe that the geological units and station term groups are qualitatively correlated. By combining the two, we propose the first frequency-dependent average site amplification functions associated with key geological units in Iceland based on age and formation. Provisionally we refer to them as hard rock, rock, (Holocene) lava rock, and (unspecified) soil site classes, respectively. We then update the GMM by incorporating these amplification functions so that it now enables the prediction of PGA and PSA at these four site classes. As a result, the new model standard deviations are considerably lower, with those associated with the event term being τ = 0.05, station terms φS2S = 0.10, intra-event φ = 0.17 and total σ = 0.18 in common logarithm units for PGA. The new site class-specific GMM now enables the site-specific ground motion for separate geological units which in turn improves site-specific seismic hazard assessment and informed site-specific engineering decision making in Iceland.

Bayesian Multiple Linear Regression Model for GDP in Nepal

Gross Domestic Product (GDP) known as the pulse of economy for any country depends on multiple factors like export-import, inflation rate and unemployment rate etc. Statistical assessment of GDP demands fresh concepts to explain GDP through its covariates in order to improve and strengthen estimation process. Descriptive analysis for the considered data set from world bank for GDP and covariates is presented through Heatmaps. Identification and relevance of possible set of covariates is done by Ordinary Least Square (OLS) regression followed by Step-wise regression. We propose an alternative statistical algorithm implemented as Bayesian Inference through Integrated Nested Laplace Approximation (INLA) which bridges the gap of accuracy in estimates as opposed to frequentist OLS regression for explaining GDP of country Nepal. Effect of changing prior parameters is assessed through Deviance Information Criterion (DIC). Different scenarios for prior distribution for regression parameters were analyzed to identify most suitable choice of parameter for normal distribution. The comparison of Bayesian and frequentist modelling results is done using several criteria such as Mean Square Error (MSE), Mean Absolute Deviation (MAD) and Mean Absolute Percent Error (MAPE). Bayesian estimation approach is a more efficient method for parametric estimation as compared to OLS classical method. GDP of Nepal was found to have strongest relationship with unemployment rate of Nepal as evident from both classical and Bayesian model.

Uncertainty quantification in mechanistic epidemic models via cross-entropy approximate Bayesian computation.

This paper proposes a data-driven approximate Bayesian computation framework for parameter estimation and uncertainty quantification of epidemic models, which incorporates two novelties: (i) the identification of the initial conditions by using plausible dynamic states that are compatible with observational data; (ii) learning of an informative prior distribution for the model parameters via the cross-entropy method. The new methodology's effectiveness is illustrated with the aid of actual data from the COVID-19 epidemic in Rio de Janeiro city in Brazil, employing an ordinary differential equation-based model with a generalized SEIR mechanistic structure that includes time-dependent transmission rate, asymptomatics, and hospitalizations. A minimization problem with two cost terms (number of hospitalizations and deaths) is formulated, and twelve parameters are identified. The calibrated model provides a consistent description of the available data, able to extrapolate forecasts over a few weeks, making the proposed methodology very appealing for real-time epidemic modeling.

An improved Hammerstein system identification method using Stein Variational Inference and sampling technology

This paper considers the identification of the Hammerstein system with immeasurable process noise. The complexity of the Hammerstein system makes it difficult to obtain accurate mathematical expressions of the parameters, or even impossible to obtain accurate mathematical expressions at all. In this contribution, we cast the Hammerstein system parameter identification problem as a posterior parameter estimation problem and take a sampling and Stein variational inference viewpoint to solve it. Improved Stein variational gradient descent(ISVGD)algorithm is proposed in posterior parameter calculation. Compared with other methods, not only the prior distribution of parameters but also the proposal distribution of parameters is considered. In other words, The posterior information is enriched and the parameter identification accuracy is improved. At the same time, ISVGD and reversible jump markov chain monte carlo (RJMCMC) algorithms are used (called ISVGD-RJMCMC algorithm) in the structure detection problem. In the algorithm, the correct basis function number k can be found in the parameter estimation. It tends to be accurate but is slow to converge. Three simulation examples were given to demonstrate the proposed algorithm’s effectiveness. Furthermore, the performances of these approaches were analyzed, including parameter estimation accuracy and error, system order estimation and parameter convergence analysis.