Reliability Analysis under Integrated Input Variable and Metamodel Uncertainty Based on Bayesian Approach

Bayesian model updating provides a rigorous framework to account for uncertainty induced by lack of knowledge about engineering systems in their respective mathematical models through updates of the joint probability density function (PDF), the so-called posterior PDF, of the unknown model parameters. The Markov chain Monte Carlo (MCMC) methods are currently the most popular approaches for generating samples from the posterior PDF. However, these methods often found wanting when sampling from difficult distributions (e.g., high-dimensional PDFs, PDFs with flat manifolds, multimodal PDFs, and very peaked PDFs). This paper introduces a new multi-level sampling approach for Bayesian model updating, called Sequential Gauss-Newton algorithm, which is inspired by the Transitional Markov chain Monte Carlo (TMCMC) algorithm. The Sequential Gauss-Newton algorithm improves two aspects of TMCMC to make an efficient and effective MCMC algorithm for drawing samples from difficult posterior PDFs. First, the statistical efficiency of the algorithm is enhanced by use of the systematic resampling scheme. Second, a new MCMC algorithm, called Gauss-Newton MCMC algorithm, is proposed which is essentially an M-H algorithm with a Gaussian proposal PDF tailored to the posterior PDF using the gradient and Hessian information of the negative log posterior. The effectiveness of the proposed algorithm for solving the Bayesian model updating problem is illustrated using three examples with irregularly shaped posterior PDFs.

Traditional full-waveform inversion (FWI) methods only render a “best-fit” model that cannot account for uncertainties of the ill-posed inverse problem. Additionally, local optimization-based FWI methods cannot always converge to a geologically meaningful solution unless the inversion starts with an accurate background model. We seek the solution for FWI in the Bayesian inference framework to address those two issues. In Bayesian inference, the model space is directly probed by sampling methods such that we obtain a reliable uncertainty appraisal, determine optimal models, and avoid entrapment in a small local region of the model space. The solution of such a statistical inverse method is completely described by the posterior distribution, which quantifies the distributions for parameters and inversion uncertainties. To efficiently sample the posterior distribution, we introduce a sampling algorithm in which the proposal distribution is constructed by the local gradient and the diagonal approximate Hessian of the local log posterior. Our algorithm is called the gradient-based Markov chain Monte Carlo (GMCMC) method. The GMCMC FWI method can quantify inversion uncertainties with estimated posterior distribution given sufficiently long Markov chains. By directly sampling the posterior distribution, we obtain a global view of the model space. Theoretically speaking, statistical assessments do not depend on starting models. Our method is applied to the 2D Marmousi model with the frequency-domain FWI setting. Numerical results suggest that our method can be readily applied to 2D cases with affordable computational efforts.

Airport air traffic is one of the most important and hardest one among all the airport data forecasts. In this paper, the Markov Chain Monte Carlo (MCMC) method of applied theory of statistics has been introduced into the aviation sector, and the discussion on airport air traffic forecast has been conducted taking Shanghai airport as the application background. MCMC method is designed to solve the problem parameters expectations where some baysian inference can not be directly calculated. This paper conducts the iterative computation using WinBUGS, EViews is used to estimate the initial iteration parameter of the regression model, and it is on this basis that the regression equation under the MCMC method is obtained.

The Bayesian approach to inverse problems, in which the posterior probability distribution on an unknown field is sampled for the purposes of computing posterior expectations of quantities of interest, is starting to become computationally feasible for partial differential equation (PDE) inverse problems. Balancing the sources of error arising from finite-dimensional approximation of the unknown field, the PDE forward solution map and the sampling of the probability space under the posterior distribution are essential for the design of efficient computational Bayesian methods for PDE inverse problems. We study Bayesian inversion for a model elliptic PDE with an unknown diffusion coefficient. We provide complexity analyses of several Markov chain Monte Carlo (MCMC) methods for the efficient numerical evaluation of expectations under the Bayesian posterior distribution, given data δ. Particular attention is given to bounds on the overall work required to achieve a prescribed error level ε. Specifically, we first bound the computational complexity of ‘plain’ MCMC, based on combining MCMC sampling with linear complexity multi-level solvers for elliptic PDE. Our (new) work versus accuracy bounds show that the complexity of this approach can be quite prohibitive. Two strategies for reducing the computational complexity are then proposed and analyzed: first, a sparse, parametric and deterministic generalized polynomial chaos (gpc) ‘surrogate’ representation of the forward response map of the PDE over the entire parameter space, and, second, a novel multi-level Markov chain Monte Carlo strategy which utilizes sampling from a multi-level discretization of the posterior and the forward PDE. For both of these strategies, we derive asymptotic bounds on work versus accuracy, and hence asymptotic bounds on the computational complexity of the algorithms. In particular, we provide sufficient conditions on the regularity of the unknown coefficients of the PDE and on the approximation methods used, in order for the accelerations of MCMC resulting from these strategies to lead to complexity reductions over ‘plain’ MCMC algorithms for the Bayesian inversion of PDEs.

The Bayesian framework is commonly used to quantify uncertainty in seismic inversion. To perform Bayesian inference, Markov chain Monte Carlo (MCMC) algorithms are regarded as the gold standard technique for sampling from the posterior probability distribution. Consistent MCMC methods have trouble for complex, high-dimensional models, and most methods scale poorly to large datasets, such as those arising in seismic inversion. As an alternative, approximate MCMC methods based on unadjusted Langevin dynamics offer scalability and more rapid sampling at the cost of biased inference. However, when assessing the quality of approximate MCMC samples for characterizing the posterior distribution, most diagnostics fail to account for these biases. In this work, we introduce the kernel Stein discrepancy (KSD) as a diagnostic tool to determine the convergence of MCMC samples for Bayesian seismic inversion. We demonstrate the use of the KSD for measuring sample quality and selecting the optimal Langevin MCMC algorithm for two Gaussian Bayesian inference problems.

Markov chain Monte Carlo (MCMC) methods offer a rapid parametric approach that can test for linkage throughout the entire genome. It has an advantage similar to nonparametric methods in that the model does not have to be completely specified a priori. However, unlike nonparametric methods, there are no limitations on pedigree size and MCMC methods can also handle relatively complex pedigree structures. In addition MCMC methods can be used to carry segregation analysis in order to answer questions on the genetic components of a disease phenotype. Segregation analysis gave evidence for between two and eight alcoholism susceptibility loci, each having a modest effect on the phenotype. MCMC methods were used to map alcoholism loci using the phenotypes ALDX1 (DSM-III-R and Feighner criteria) and ALDX2 (World Health Organization diagnosis ICD-10 criteria). There was mild evidence for quantitative trait loci on chromosomes 2, 10, and 11.

Knowledge of the Earth's interior relies on indirect information collected at or near the surface. Typically, data do not uniquely constrain the subsurface properties and are contaminated by noise, and therefore the solution of ill-posed inverse problems is required. Geophysicists have traditionally addressed such problems through deterministic approaches, seeking a single 'best-fitting' model defined by optimality criteria that reflect our understanding of the problem at hand. In recent decades, Bayesian approaches have become increasingly common, as they characterise the posterior probability distribution of the model conditioned on the observations, thereby quantifying uncertainty.In this context, Markov chain Monte Carlo (MCMC) methods have emerged as a crucial tool as they allow sampling from posterior distributions of arbitrary complexity. At the core of many MCMC algorithms lies the Metropolis-Hastings scheme. This combines a proposal distribution with a probabilistic acceptance criterion to construct a Markov chain that has the desired target distribution as its stationary distribution. The algorithm is versatile as it rests on mild technical conditions on the proposal, and is thus widely adopted across a broad range of geoscientific inference problems. Yet when the parameter space is large or the forward models are computationally expensive---both common scenarios in geophysical applications---it can become inefficient, resulting in poor chain mixing and slow convergence to the target (posterior) distribution. These challenges underscore the importance of effective proposal mechanisms.In this presentation, we introduce a novel approach to designing Metropolis-Hastings proposals based on adaptive transport maps. The framework is inspired by recent developments from the field of Applied Mathematics linking Bayesian inference with Optimal Transport theory. The idea is to find a monotone, nonlinear transformation to recast a (complex) target probability distribution into a (simpler) reference distribution that is more amenable to standard MCMC steps. Our key contribution is to parameterise these transformations using invertible neural networks, ensuring monotonicity while gaining the flexibility and expressiveness that neural architectures afford.The proposed method progresses iteratively. We begin with a standard sampling strategy (e.g., a random-walk Metropolis) to obtain initial draws from the target distribution. These samples inform the training of an invertible neural network that learns to map from the target to a simpler reference distribution, specifically a standard (multivariate) Gaussian. Subsequent proposals are then generated in the reference space, either as global independence moves or local perturbations, and are accepted or rejected following a suitably modified Metropolis-Hastings criterion. As more samples accumulate, the network's parameters are updated, improving overall sampling efficiency. Ultimately, the approach yields not only an ensemble of samples representative of the desired target distribution---just as in standard MCMC---but also a compact, learned representation of it in the network's weights.We illustrate the proposed paradigm both theoretically and through examples. Preliminary results indicate that transport-map-enhanced MCMC has the potential to significantly accelerate Bayesian sampling across a range of applications.

Positron Emission Tomography (PET) is a valuable imaging method for studying molecular-level processes in the body, such as hyperphosphorylated tau (p-tau) protein aggregates, a hallmark of several neurodegenerative diseases including Alzheimer's disease. P-tau density and cerebral perfusion can be quantified from PET data using tracer kinetic modeling techniques. However, noise in PET images leads to uncertainty in the estimated kinetic parameters. This can be quantified in a Bayesian framework by the posterior distribution of kinetic parameters given PET measurements. Markov Chain Monte Carlo (MCMC) techniques can be employed to estimate the posterior distribution, although with significant computational needs. In this paper, we propose to leverage deep learning inference efficiency to infer the posterior distribution. A novel approach using denoising diffusion probabilistic model (DDPM) is introduced. The performance of the proposed method was evaluated on a [18F]MK6240 study and compared to an MCMC method. Our approach offered significant reduction in computation time (over 30 times faster than MCMC) and consistently predicted accurate (< 0.8 % mean error) and precise (< 5.77 % standard deviation error) posterior distributions.

The reliability analysis methods based on the Kriging model have been explored to update the design of experiments (DoE), but the number of calls to the performance function is high. In this paper, a new active learning method combining the weight information entropy function (WH) and the adaptive candidate sample pool is proposed to improve the reliability analysis efficiency. Based on the information entropy function (H), the learning function WH is constructed to consider Kriging variance and the joint probability density function (PDF). In this proposed method, the sample points can be assigned different weight by the learning function WH according to the important degrees of sample points. The point which not only nears the limit state function (LSF), but also has a high probability density function value and large Kriging variance is assigned more weight than others. To select the sample points with lower confidence level, the adaptive candidate sample pool is generated by Markov Chain Monte Carlo (MCMC) simulation. The Kriging model can be updated efficiently by the proposed method. Four numerical examples and an engineering example with implicit performance function are used to verify the efficiency and accuracy of the proposed method. The results show that the proposed method can significantly improve the computational efficiency of the reliability analysis without losing accuracy.

Identification of contaminant sources in rivers is crucial for river protection and emergency response. This study presents an innovative approach for identifying river pollution sources by using Bayesian inference and cellular automata (CA) modeling. A general Bayesian framework is proposed that combines the CA model with observed data to identify unknown sources of river pollution. To reduce the computational burden of the Bayesian inference, a CA contaminant transport model is developed to efficiently simulate pollutant concentration values in the river. These simulated concentration values are then used to calculate the likelihood function of available measurements. The Markov chain Monte Carlo (MCMC) method is used to produce the posterior distribution of contaminant source parameters, which is a sampling-based method that enables the estimation of complex posterior distributions. The suggested methodology is applied to a real case study of the Fen River in Yuncheng City, Shanxi Province, Northern China, and it estimates the release time, release mass, and source location with relative errors below 19%. The research indicates that the proposed methodology is an effective and flexible way to identify the location and concentrations of river contaminant sources.

The Guide to the Expression of Uncertainty in Measurement (GUM) has led to a harmonization of uncertainty evaluation throughout metrology. The simplicity of its employed methodology has fostered the broad acceptance of the GUM among metrologists. However, this simplicity also compromises best practice and does not provide state-of-the-art data analysis. Specifically, metrologists often possess useful prior knowledge about the measurand which cannot be accounted for by the GUM. Bayesian uncertainty evaluation, on the other hand, takes prior knowledge about the measurand as its starting point. While this technique has been successfully applied in several instances in metrology, its broad access for metrologists is still lacking. One reason for this is that application of Bayesian methods and their computation, e.g. by means of Markov chain Monte Carlo (MCMC) methods, usually requires statistical skills.In this work, we propose a simple method for Bayesian uncertainty evaluation applicable to measurement models that depend linearly on a single input quantity for which Type A information is available, and several input quantities for which Type B information is given. This category of measurement models covers many uncertainty evaluations in metrology. The approach utilizes a specific class of prior distributions to encode the prior information about the measurand. Explicit guidance is provided on how to choose the parameters of the prior in the light of one’s prior knowledge. MCMC methods are not required for the calculation of results; instead, a simple Monte Carlo procedure is developed that allows to draw uncorrelated samples from the posterior distribution, which greatly simplifies convergence assessments. The proposed Bayesian approach allows the treatment of small numbers of observations for a Type A uncertainty evaluation, including the case of only a single observation. Examples are provided that illustrate the proposed approach and corresponding open source Python software is made available.

In this paper, the authors outline the general principles behind an approach to Bayesian system identification and highlight the benefits of adopting a Bayesian framework when attempting to identify models of nonlinear dynamical systems in the presence of uncertainty. It is then described how, through a summary of some key algorithms, many of the potential difficulties associated with a Bayesian approach can be overcome through the use of Markov chain Monte Carlo (MCMC) methods. The paper concludes with a case study, where an MCMC algorithm is used to facilitate the Bayesian system identification of a nonlinear dynamical system from experimentally observed acceleration time histories.

The maximum likelihood and Bayesian approaches for parameter estimations and prediction of future record values have been considered for the two-parameter Burr Type XII distribution based on record values with the number of trials following the record values (inter-record times). Firstly, the Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, the Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. Secondly, the Bayes estimates are obtained with respect to a discrete prior for the first shape parameter and a conjugate prior for other shape parameter. The Bayes and the maximum likelihood estimates are compared in terms of the estimated risk by the Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record arising from the Burr Type XII distribution based on record data. The comparison of the derived predictors is carried out by using Monte Carlo simulations. A real data are analysed for illustration purposes.

We describe a Bayesian approach to estimating luminosity functions. We derive the likelihood function and posterior probability distribution for the luminosity function, given the observed data, and we compare the Bayesian approach with maximum-likelihood by simulating sources from a Schechter function. For our simulations confidence intervals derived from bootstrapping the maximum-likelihood estimate can be too narrow, while confidence intervals derived from the Bayesian approach are valid. We develop our statistical approach for a flexible model where the luminosity function is modeled as a mixture of Gaussian functions. Statistical inference is performed using Markov chain Monte Carlo (MCMC) methods, and we describe a Metropolis-Hastings algorithm to perform the MCMC. The MCMC simulates random draws from the probability distribution of the luminosity function parameters, given the data, and we use a simulated data set to show how these random draws may be used to estimate the probability distribution for the luminosity function. In addition, we show how the MCMC output may be used to estimate the probability distribution of any quantities derived from the luminosity function, such as the peak in the space density of quasars. The Bayesian method we develop has the advantage that it is able to place accurate constraints on the luminosity function even beyond the survey detection limits, and that it provides a natural way of estimating the probability distribution of any quantities derived from the luminosity function, including those that rely on information beyond the survey detection limits.

Stochastic parameter estimation and inversion have become increasingly popular in recent years. Nowadays, it is computationally reasonable and regular to solve complex inverse problems within the Bayesian framework. Applications of Bayesian inferences for inverse problems require investigation of the posterior distribution, which usually has a complex landscape and is highly dimensional. In these cases, Markov chain Monte Carlo methods (MCMC) are often used. This paper discusses a Bayesian approach for identifying adsorption and desorption rates in combination with a pore-scale reactive flow. Markov chain Monte Carlo sampling is used to estimate adsorption and desorption rates. The reactive transport in porous media is governed by incompressible Stokes equations, coupled with convection–diffusion equation for species’ transport. Adsorption and desorption are accounted via Robin boundary conditions. The Henry isotherm is considered for describing the reaction terms. The measured concentration at the outlet boundary is provided as additional information for the identification procedure. Metropolis–Hastings and Adaptive Metropolis algorithms are implemented. Credible intervals have been plotted from sampled posterior distributions for both algorithms. The impact of the noise in the measurements and influence of several measurements for Bayesian identification procedure is studied. Sample analysis using the autocorrelation function and acceptance rate is performed to estimate mixing of the Markov chain. As result, we conclude that MCMC sampling algorithm within the Bayesian framework is good enough to determine an admissible set of parameters via credible intervals.