Posterior Distribution Of Parameters Research Articles

A two-stage Bayesian model updating framework based on an iterative model reduction technique using modal responses

A two-stage Bayesian model updating framework based on an affine-invariance sampling in Transitional Markov Chain Monte Carlo (TMCMC) algorithm is proposed. In the first stage, unknown modal coordinates are identified with an improved iterative model reduction technique. Subsequently, a modified TMCMC algorithm is proposed where obtaining the statistical estimators in the usual TMCMC algorithm is not necessary. In detail, an affine-invariance sampling approach is proposed to estimate a multi-dimensional scaling (MDS) factor in the affine-transformed space that accounts for the frequencies and mode shapes which are of different natures and dimensions at the levels of likelihood, prior and posterior distributions. The TMCMC sampling adaptively utilizes the plausibility value from the proposed MDS-based tuning algorithm to improve its transition levels. The proposed algorithm in the affine-invariance sampling space accounting for the observables of different dimensions is expected to facilitate the estimation of the posterior distribution of model parameters. The effectiveness of the proposed algorithm is demonstrated by considering a simply-supported steel beam, a ten-storied reinforced concrete building model and an eight-degree-of-freedom spring–mass model for which experimental data is available. The second example demonstrates the applicability of the approach for updating large finite element models involving substructuring.

Proxy-based Bayesian inversion of strain tensor data measured during well tests

Recent instrument developments have made it possible to measure the strain tensor caused by injecting or pumping fluid from aquifers or reservoirs, but the full value of these data is limited because the long runtimes of poroelastic forward models makes it impractical to use many inversion schemes. This limits the interpretation of strain data for managing the recovery of resources or storage of wastes in the subsurface. This paper describes a method of inverting deformation data using a poroelastic numerical simulator so the results can be used to manage reservoirs or aquifers. We developed a workflow designed to reduce the number of simulations sufficiently to make it feasible to use DREAMzs, an advanced Bayesian inversion method that translates the uncertainties from different sources into unbiased posterior parameter distributions and uncertainty envelopes around the field data. Using a KNN proxy model for the poroelastic simulator is key to reducing the overall computations, and the workflow includes a strategy for ensuring the proxy model results converge on the results from the simulator. The workflow is tested using an idealized example that verifies the ability to correctly identify parameters and characterize noise used to perturb the data. Field data from an injection test at an oil reservoir near Tulsa, Oklahoma, are also used to evaluate the efficacy of the workflow with a real dataset. The workflow identified 265 history matching solutions out of 1240 total simulation runs (21% acceptance ratio), where the results were used to characterize posterior parameter distribution and evaluate the prediction uncertainty. This workflow is significant because it enables strain tensor, or other geomechanical measurements to be interpreted to guide decision-making during energy and environmental processes in the subsurface.

Effect existence of aging on stutter ratio evaluated via Bayesian inference

The stochastic behavior of the stutter ratio (SR) in capillary electrophoresis-based DNA typing is currently described and predicted using statistical models in forensic genetics. Clarifying this behavior can help obtain more objective and robust evidence to the court in terms of mixture interpretation. This study aimed to investigate the effect existence of aging on SR via a Bayesian framework. Nail scrapings and clippings were collected from 68 healthy individuals with informed consent. Samples were classified by age-class: young group (0–16 years; n = 36) and older-adult group (>61 years; n = 32). Then, they were compared in terms of their SRs for each simple repeat locus included in GlobalFiler Kit. Bayesian modeling was performed with lognormal distribution model, which implemented multiple linear regression, allele and age-class as explanatory variables. For all simple repeat loci, the median of the posterior distribution of the age-class parameter was a positive value. For CSF1PO and D7S820, the 95% credible interval of the posterior distribution did not include 0. Our data suggested that aging slightly increases the SR. These findings might help elucidate the stochastic behavior of SR.

Investigation of half-normal model using informative priors under Bayesian structure

This paper considers properties of half-normal distribution using informative priors under the Bayesian criterion. It employs the squared root inverted gamma, Chi-square and Rayleigh distributions as the prior distribution to construct the Posterior distributions of the respective distributional parameters. Hyperparameters are elicited via prior predictive distribution. The properties of posterior distribution are studied, and their graphs are presented using a real data set. A comprehensive simulation scheme is conducted using informative priors. Bayes estimates are obtained using the loss functions (squared error loss function, modified loss function, quadratic loss function and Degroot loss function). Statistical inferences interval estimates and Bayesian hypothesis testing are presented to demonstrate the usefulness of the study.

Bayesian Identification Procedure for Triple Seasonal Autoregressive Models

Triple seasonal autoregressive (TSAR) models have been introduced to model time series date with three layers of seasonality; however, the Bayesian identification problem of these models has not been tackled in the literature. Therefore, in this paper, we have the objective of filling this gap by presenting a Bayesian procedure to identify the best order of TSAR models. Assuming that the TSAR model errors are normally distributed along with employing three priors, i.e., normal-gamma, Jeffreys’ and g priors, on the model parameters, we derive the marginal posterior distributions of the TSAR model parameters. In particular, we show that the marginal posteriors are multivariate t and gamma distributions for the TSAR model coefficients vector and precision, respectively. Using the marginal posterior distribution of the TSAR model coefficients vector, we present an identification procedure for the TSAR models based on a sequence of t-test of significance. We evaluate the accuracy of the proposed Bayesian identification procedure by conducting an extensive simulation study, followed by a real application to hourly electricity load datasets in six European countries.

A recursive inference method based on invertible neural network for multi-level model updating using video monitoring data

Likelihood-free inference methods have been widely adopted, but they face significant challenges in updating multi-level computational models that have hierarchically embedded sub-models. This difficulty arises from the lack of direct observations of the quantities of interest of the sub-models. In addition, recent advancements in sensing and image processing technologies allow for the collection of a substantial amount of video monitoring data through non-contact sensing techniques. The implicit and very complicated relationship between the uncertain model parameters and video monitoring data adds an additional layer of challenge to the updating of multi-level models. This paper overcomes these challenges by proposing an innovative Recursive Inference method based on Invertible Neural Networks (RINN) for multi-level models. The proposed RINN framework first compresses the high-dimensional video monitoring data into low-dimension latent-space data using a convolutional autoencoder. Using synthetic video monitoring data generated by considering various uncertainty sources in the computational simulation models, a likelihood-free inference model is then trained through a conditional invertible neural network (cINN) and a summary neural network. This model efficiently approximates the posterior distributions of uncertain model parameters without evaluating the computationally intractable likelihood function, given any latent-space data obtained from the autoencoder. To facilitate continuous monitoring and model updating over an extended monitoring period, this paper further proposes a recursive model updating strategy that integrates the cINN-based likelihood-free inference with the particle filtering method. The updating of a degradation model of a miter gate application is employed as an example throughout this paper to explain and demonstrate the efficacy of the proposed RINN framework. The results of the case study show that the RINN is able to effectively reduce uncertainty in the degradation model parameters through strain video monitoring data of a miter gate, and thereby increase the confidence in the remaining useful life (RUL) estimation using the updated degradation model.

Asteroseismology of δ Scuti stars: emulating model grids using a neural network

ABSTRACT Young δ Scuti (Sct) stars have proven to be valuable asteroseismic targets, but obtaining robust uncertainties on their inferred properties is challenging. We aim to quantify the random uncertainties in grid-based modelling of δ Sct stars. We apply Bayesian inference using nested sampling and a neural network emulator of stellar models, testing our method on both simulated and real stars. Based on results from simulated stars, we demonstrate that our method can recover plausible posterior probability density estimates while accounting for both the random uncertainty from the observations and neural network emulation. We find that the posterior distributions of the fundamental parameters can be significantly non-Gaussian and multimodal, and have strong covariance. We conclude that our method reliably estimates the random uncertainty in the modelling of δ Sct stars and paves the way for the investigation and quantification of the systematic uncertainty.

Gradient boosting Bayesian neural networks via Langevin MCMC

Bayesian neural networks harness the power of Bayesian inference which provides an approach to neural learning that not only focuses on accuracy but also uncertainty quantification. Markov Chain Monte Carlo (MCMC) methods implement Bayesian inference by sampling from the posterior distribution of the model parameters. In the case of Bayesian neural networks, the model parameters refer to weights and biases. MCMC methods suffer from scalability issues in large models, such as deep neural networks with thousands to millions of parameters. In this paper, we present a Bayesian ensemble learning framework that utilizes gradient boosting by combining multiple shallow neural networks (base learners) that are trained by MCMC sampling. We present two Bayesian gradient boosting strategies that employ simple neural networks as base learners with Langevin MCMC sampling. We evaluate the performance of these methods on various classification and time-series prediction problems. We demonstrate that the proposed framework improves the prediction accuracy of canonical gradient boosting while providing uncertainty quantification via Bayesian inference. Furthermore, we demonstrate that the respective methods scale well when the size of the dataset and model increases.

Sequential Bayesian Planning for Accelerated Degradation Tests Considering Sensor Degradation

Most classical accelerated degradation test (ADT) planning models implicitly overlook the errors when measuring the degradation levels of the test units. However, the sensor measurement errors are inevitable and the magnitude of the errors may have a trend to increase over time due to sensor degradation. As a consequence improperly overlooking the sensor degradation in ADT planning could result in a test plan with unsatisfactory performance. This article addresses this issue by proposing a sequential ADT planning model that factors in sensor degradation. The system degradation level is periodically measured, based on which we dynamically adjust the stress level during ADT. We adopt a Bayesian framework that periodically updates the posterior distribution of model parameters considering the sensor degradation. An approximate Bayesian computation algorithm is developed to circumvent the difficulty of directly evaluating the complicated likelihood function in our problem. Numerical studies on a gas turbine reveal that our sequential model outperforms several traditional ADT designs that overlook the sensor degradation.

Fundamental parameters of 318 contact binaries from the TESS survey

ABSTRACT The TESS Survey has released a large number of high-precision light curves of contact binaries. However, using the Phoebe program and Markov chain Monte Carlo (MCMC) algorithm to obtain the posterior distribution of contact binary parameters is a time-consuming process. In order to obtain the contact binary parameters from the TESS survey, we build neural network (NN) models and combine them with the MCMC algorithm to obtain the contact binary parameters and parameter errors quickly. NN model is used in place of the physical model, which can generate a light curve with a precision of less than a millimagnitude. The NN model is capable of generating light curves at a speed that is four orders of magnitude faster than Phoebe running on the same computing platform. In this study, we have determined the parameters of 318 contact binary systems exhibiting relatively symmetric light curves. Subsequently, a statistical analysis was conducted on the derived parameters of these 318 targets. The coefficient of determination (R2) for 318 contact binaries between the light curves generated by Phoebe using these parameters obtained by the NN model and MCMC as inputs and the original light curves is greater than 0.99. Additionally, the distribution and correlation of the parameters for these 318 contact binary systems have been presented.

Guided wave-based characterisation of cracks in pipes utilising approximate Bayesian computation

This paper proposed a probabilistic framework of damage characterisation to detect and identify early-state cracks in pipe-like structures using ultrasonic guided waves. The crack location, crack sizes (e.g., depth and width of the crack), and Young's modulus are considered as unknown parameters in the model updating using a Bayesian approach, by which their values and the associated uncertainties are quantified. The proposed framework is developed based on approximate Bayesian computation (ABC) by subset simulation, which is a likelihood-free Bayesian approach. This algorithm estimates the posterior distributions of unknown parameters by directly accessing the similarity between the measured signals from experiments and the simulated guided wave (GW) signals from the numerical model. In this case, the evaluation of likelihood functions can be smartly circumvented during Bayesian inference. A time-domain spectral finite element (SFE) method with a cracked finite element model is employed to model the pipes to enhance the computational efficiency of the simulation and model updating. Numerical and experimental case studies are carried out to evaluate the performance of the proposed likelihood-free approach. Numerical results show the accuracy and robustness of the proposed approach in identifying unknown parameters under different scenarios. The associated uncertainties of each parameter are also quantified by analysing the statistical properties of the sample set, such as mean and coefficient of variation (COV) values. Experimental results show that the proposed method can accurately identify the unknown parameters, which further verifies the accuracy and practicability of the probabilistic damage characterisation framework.

Validating DayCent-CR for cropland soil carbon offset reporting at a national scale

Regenerative soil management practices have been shown to increase soil organic carbon in cropland previously under conventional management, and farmers that adopt regenerative practices could be eligible to participate in carbon offset programs. Due to the high cost of soil sampling at large scales, project developers of agricultural carbon offset programs may employ a hybrid measurement and modeling approach to SOC quantification. While biogeochemical models allow for carbon crediting to occur on larger scales than soil sampling alone would allow, any model used must be unbiased and shown to adequately predict SOC changes, with known uncertainty, across the crops, practice changes, and geographies of interest. The “credit-ready” version of the DayCent ecosystem model, DayCent-CR, was evaluated for performance across 14 combinations of crops and practice categories. Model calibration and validation was performed with a Bayesian Markov chain Monte Carlo approach using k-fold cross validation and 668 SOC stock change measurements from 41 agricultural research sites. Overall model performance met the guidelines established by Climate Action Reserve’s Soil Enrichment Protocol: ≥90% of model prediction intervals covered the measured value, and mean bias in all categories was less than pooled measurement uncertainty. Importantly, posterior distributions of DayCent-CR parameters and variance components enable the calculation of variance, which can then be used to calculate an uncertainty deduction that is applied to overall project credits to ensure conservatism. The calibrated model parameters are therefore valid for use in crediting programs within the domain of the validation dataset.

Prediction of concrete carbonation based on the inverse Gaussian process and Bayesian method

Concrete carbonation is one of the major factors causing the deterioration of reinforced concrete structures. Therefore, accurately predicting the carbonation depth is of great significance in safety assessments of structures. The aim of this study was to develop a method to predict carbonation behaviour by incorporating multi-source information using the Bayesian method. First, the inverse Gaussian process was used to model the evolution of carbonation depth; this captured the temporal variability and the monotonicity of the deterioration phenomenon very well. Then, a proper prior for the model was determined using knowledge from the existing empirical carbonation model. To fuse the accelerated carbonation data and field inspection data, Bayesian inference was performed to update the posterior distributions of the model parameters by the Gibbs sampling technique. A practical example case was used to illustrate the validity and accuracy of the proposed approach.

An uncertainty quantification framework for logistic regression based geospatial natural hazard modeling

There is a class of data-driven global natural hazard predictive models that take advantage of broadly available geospatial proxies. These data-driven geospatial models have been commonly used for landslides and are becoming more available in recent years for liquefaction. Logistic regression is the most common method for predicting these ground failure occurrences. These models do not often include robust quantification of uncertainties although they are widely used in the pre-disaster planning and post-disaster response around the world. Taking the logistic regression based global geospatial liquefaction model (GGLM) (Zhu et al., 2017) as an example, we propose an uncertainty quantification (UQ) framework that consists of characterization of different sources of uncertainty, model sensitivity analysis, and forward uncertainty propagation. In this study, we have identified the main sources of uncertainty in such predictive models as parameter estimation uncertainty, modeling error, and geospatial input uncertainty. A Bayesian inference algorithm is used to quantify the posterior distribution of model parameters and quantify model parameter estimation uncertainties which are found to be negligible when a large amount of data is used in the parameter estimation process. Modeling errors are characterized based on the observed residuals between model predictions and measurements and by fitting a Gaussian distribution to the liquefaction probability residuals. The geospatial input uncertainties are characterized using the literature and expert judgment and propagated into model output. Second, we investigate the sensitivity of model output to different uncertain inputs and find that the variance of model output is largely controlled by the geospatial input uncertainties and model errors. Last, we propose an approximate forward uncertainty propagation method, which provides comparable results to a Monte Carlo simulation-based method with better computational efficiency. The proposed UQ framework provides a measure of uncertainty on model predictions and can be applied to any logistic-regression models and other geospatial modeling problems.

Stable water isotopes and tritium tracers tell the same tale: no evidence for underestimation of catchment transit times inferred by stable isotopes in StorAge Selection (SAS)-function models

Abstract. Stable isotopes (δ18O) and tritium (3H) are frequently used as tracers in environmental sciences to estimate age distributions of water. However, it has previously been argued that seasonally variable tracers, such as δ18O, generally and systematically fail to detect the tails of water age distributions and therefore substantially underestimate water ages as compared to radioactive tracers such as 3H. In this study for the Neckar River basin in central Europe and based on a >20-year record of hydrological, δ18O and 3H data, we systematically scrutinized the above postulate together with the potential role of spatial aggregation effects in exacerbating the underestimation of water ages. This was done by comparing water age distributions inferred from δ18O and 3H with a total of 21 different model implementations, including time-invariant, lumped-parameter sine-wave (SW) and convolution integral (CO) models as well as StorAge Selection (SAS)-function models (P-SAS) and integrated hydrological models in combination with SAS functions (IM-SAS). We found that, indeed, water ages inferred from δ18O with commonly used SW and CO models are with mean transit times (MTTs) of ∼ 1–2 years substantially lower than those obtained from 3H with the same models, reaching MTTs of ∼10 years. In contrast, several implementations of P-SAS and IM-SAS models not only allowed simultaneous representations of storage variations and streamflow as well as δ18O and 3H stream signals, but water ages inferred from δ18O with these models were, with MTTs of ∼ 11–17 years, also much higher and similar to those inferred from 3H, which suggested MTTs of ∼ 11–13 years. Characterized by similar parameter posterior distributions, in particular for parameters that control water age, P-SAS and IM-SAS model implementations individually constrained with δ18O or 3H observations exhibited only limited differences in the magnitudes of water ages in different parts of the models and in the temporal variability of transit time distributions (TTDs) in response to changing wetness conditions. This suggests that both tracers lead to comparable descriptions of how water is routed through the system. These findings provide evidence that allowed us to reject the hypothesis that δ18O as a tracer generally and systematically “cannot see water older than about 4 years” and that it truncates the corresponding tails in water age distributions, leading to underestimations of water ages. Instead, our results provide evidence for a broad equivalence of δ18O and 3H as age tracers for systems characterized by MTTs of at least 15–20 years. The question to which degree aggregation of spatial heterogeneity can further adversely affect estimates of water ages remains unresolved as the lumped and distributed implementations of the IM-SAS model provided inconclusive results. Overall, this study demonstrates that previously reported underestimations of water ages are most likely not a result of the use of δ18O or other seasonally variable tracers per se. Rather, these underestimations can largely be attributed to choices of model approaches and complexity not considering transient hydrological conditions next to tracer aspects. Given the additional vulnerability of time-invariant, lumped SW and CO model approaches in combination with δ18O to substantially underestimate water ages due to spatial aggregation and potentially other still unknown effects, we therefore advocate avoiding the use of this model type in combination with seasonally variable tracers if possible and instead adopting SAS-based models or time-variant formulations of CO models.

Quantification of material modeling uncertainty on seismic performance of SMA-based self-centering concentrically braced frames

Shape memory alloys (SMAs) combine super-elasticity and energy dissipation capacity therefore is favored in seismic hazard mitigation. Simplifications are necessary to incorporate acquired material properties through material testing for dynamic analysis. This study presents a methodology to integrate SMA material testing, model inference, and probabilistic seismic analysis for uncertainty quantification of SMA model uncertainty on SMA based structures. The proposed methodology is applied to a SMA-based self-centering concentrically braced frame (SC-CBF). The posterior distribution of SMA modeling parameters is first obtained through the MCMC algorithm. Considering computational cost of MCMC algorithm, linear moments method is introduced to sample SMA modeling parameters. Extensive time-history analysis is then performed to quantify the effect of SMA modelling uncertainty and ground motions on the performance of SC-CBF. Significant response changes can be observed in SC-CBF due to the effect of uncertainty, which inevitably affect the reliability assessment of the structure. These analysis results also show that modeling uncertainty will affect seismic performance of SMA-based SC-CBF but its influence is less significant than that caused by ground motion uncertainty for the braced frame investigated in this study.

Bayesian Analysis of Change Point Problems Using Conditionally Specified Priors

In data analysis, change point problems correspond to abrupt changes in stochastic mechanisms generating data. The detection of change points is a relevant problem in the analysis and prediction of time series. In this paper, we consider a class of conjugate prior distributions obtained from conditional specification methodology for solving this problem. We illustrate the application of such distributions in Bayesian change point detection analysis with Poisson processes. We obtain the posterior distribution of model parameters using general bivariate distribution with gamma conditionals. Simulation from the posterior are readily implemented using a Gibbs sampling algorithm. The Gibbs sampling is implemented even when using conditional densities that are incompatible or only compatible with an improper joint density. The application of such methods will be demonstrated using examples of simulated and real data.

A Bayesian method with nonlinear noise model to calibrate constitutive parameters of soft tissue

The measured mechanical responses of soft tissue exhibit large variability and errors, especially for the softest brain tissue, while calibrating its constitutive parameters in a deterministic way remains a common practice. Here we implement a Bayesian method considering the nonlinear noise model to calibrate constitutive parameters of brain tissue. A probability model is first developed based on the measured experimental data, likelihood function, and prior function, from which the posterior distributions of model parameters are formulated. The likelihood function considers the nonlinear behaviors of the constitutive response and noise distribution of the experimentally measured data. Meanwhile, the prior predictive distribution is computed to check the probability model preliminarily. Secondly, the Markov Chain Monte Carlo (MCMC) method is used to compute the posterior distributions of model parameters, enabling assessment of parameter uncertainty, correlation, and model calibration error. Finally, the posterior predictive distributions of the overall response, constitutive response, and noise response are computed to validate the probabilistic model, all of which are consistent with the corresponding data. Furthermore, the effect of the prior distribution, experimental data, and noise model on posterior distribution is studied. Our study provides a general approach to calibrating constitutive parameters of soft tissue despite errors and large variability in experimental data.

Statistical inference of reliability distributions based on progressively hybrid censoring data

This paper studies the statistical inferences of the unknown parameters of some reliability models using progressively hybrid censoring of type-I and type-II. We apply the maximum likelihood and Bayesian approaches to perform the statistical inferences of the parameters when the data follow Weibull, power Lindley, and Sarhan–Tadj–Hamilton distributions. Bayes method is implemented under the squared error loss function when the parameters are independent and follow gamma prior distributions with known hyperparameters. The maximum likelihood estimates cannot be obtained in an analytic form. Therefore we use the R function “optim” to find those estimates. Also, the joint posterior distribution of the unknown parameters cannot be obtained in closed form. Therefore, we will use the Markov Chain Monte Carlo method to approximate the Bayesian analysis. We constructed the highest posterior density intervals of the model parameters. We analyzed three real data sets for illustration and comparison purposes. We found out that Sarhan–Tadj–Hamilton distribution fits those three real data sets better than Weibull and Power Lindley distribution using both complete and progressively hybrid censored samples. We used expected experimentation time to discuss the optimal test plans. Simulation studies are performed to investigate the proposed methods. Based on the simulation results, we could conclude that Bayesian method does not provide better estimations than the maximum likelihood method.

A novel variational robust filter with Gaussian mixture model for unknown non-Gaussian noises

In numerous practical applications, the presence of unpredictable abnormal behaviors often leads to random measurement outliers, causing the random measurement noises are no longer Gaussian but unknown non-Gaussian distributions, and it is intractable to obtain accurate a priori information about the noises. To address this challenge and enable nonlinear state estimation under such measurement noises, we propose a novel robust filter based on Gaussian mixture model (GMM). Firstly, a GMM is introduced to model the unknown non-Gaussian measurement likelihood probability density function (ML-PDF). Without relying on any prior noise statistics, the GMM can accurately model the ML-PDF utilizing a set of model parameters adaptively learned by the variational Bayesian (VB) method. Then, considering the high computational burden associated with the existing variational iterative process, we develop an improved VB method, which independently calculates the posterior distributions of state and unknown model parameters, thereby improving the computational efficiency. Finally, a new robust filter is derived by integrating the GMM with the improved VB approach, The effectiveness and superiority of the filter are demonstrated through both numerical simulations and real-world data analysis.