Conventional Markov Chain Monte Carlo Research Articles

2-D probabilistic inversion of MT data and uncertainty quantification using the Hamiltonian Monte Carlo method

SUMMARY Bayesian methods provide a valuable framework for rigorously quantifying the model uncertainty arising from the inherent non-uniqueness in the magnetotelluric (MT) inversion. However, widely used Markov chain Monte Carlo (MCMC) sampling approaches usually require a significant number of model samples for accurate uncertainty estimates, making their applications computationally challenging for 2-D or 3-D MT problems. In this study, we explore the applicability of the Hamiltonian Monte Carlo (HMC) method for 2-D probabilistic MT inversion. The HMC provides a mechanism for efficient exploration in high-dimensional model space by making use of gradient information of the posterior probability distribution, resulting in a substantial reduction in the number of samples needed for reliable uncertainty quantification compared to the conventional MCMC methods. Numerical examples with synthetic data demonstrate that the HMC method achieves rapid convergence to the posterior probability distribution of model parameters with a limited number of model samples, indicating the computational advantages of the HMC in high-dimensional model space. Finally, we applied the developed approach to the COPROD2 field data set. The statistical models derived from the HMC approach agree well with previous results obtained by 2-D deterministic inversions. Most importantly, the probabilistic inversion provides valuable quantitative model uncertainty information associated with the resistivity structures derived from the observed data, which facilitates model interpretation.

Scalable and portable computational framework enabling online probabilistic remaining useful life (RUL) estimation

This work demonstrates a framework that enables online prognostics in adhesive joints by estimating the real-time probabilistic remaining useful life (RUL) using ANNs based hybrid physics models and vectorized sequential Monte Carlo (SMC) simulations. The framework is developed by integrating the physics-based damage degradation model and uncertainty quantification (UQ) techniques to estimate both probabilistic fatigue failure life and RUL. The fatigue damage growth (FDG) simulator, a hybrid surrogate model that simulates real-time fatigue degradation in adhesive joints, is used. In the initial set of results, the generalized framework is validated by estimating the probabilistic fatigue failure life using two UQ methods: Markov Chain Monte Carlo (MCMC) and SMC method. The computational results are successfully compared against experimental data. The conventional MCMC sampling methods are inherently serial, which limits the exploitation of the computational speed-up provided by the FDG simulator and hinders the real-time life predictions. The SMC method quantifies the uncertainties by parallelizing the sampling process, significantly reducing computational time and enabling real-time prediction. Next, the generalized framework is used to estimate probabilistic RUL from the fatigue crack propagation data. The parallel SMC method showed very good speedup compared to the MCMC method. To further enhance the computational speed-up with SMC method, vectorized FDG simulations are introduced into the framework and good scalability is achieved. Finally, the portability of the framework is demonstrated by deploying it on the portable Raspberry Pi cluster.

Surrogate modeling for Bayesian inverse problems based on physics-informed neural networks

The fundamental computing issues in Bayesian inverse problems (BIPs) stem from the need for repeated forward model evaluations, which are required by Markov chain Monte Carlo (MCMC) methods for posterior sampling. In this paper, we present an offline-online computational strategy for combining conventional MCMC with physics-informed neural networks (PINNs) for BIPs to achieve gains that cannot be reached with either component alone. In other words, we develop an offline learning technique that employs PINNs to construct a surrogate of the forward model, which we then utilize to build a new simulator for online sampling. During the online stage, the surrogate model is efficiently fine-tuned based on the current sample position, ensuring numerical accuracy in local regions. We specifically developed two types of online PINNs-based algorithms: transfer learning PINNs (TPINNs) and multi-fidelity PINNs (MPINNs). The proposed methods can obtain accurate posterior information with a small number of forward simulations, as demonstrated by numerical examples.

A Novel Method for Damage Identification Based on Tuning-Free Strategy and Simple Population Metropolis–Hastings Algorithm

The most commonly used method for sampling damage parameters from the posterior distribution is the Markov chain Monte Carlo (MCMC) method. The population MCMC method as one of the MCMC methods has been utilized for damage identification by some researchers recently. Nevertheless, for the conventional population MCMC methods, these sampling methods often require significant computational resources and tuning of a large number of algorithm parameters. Aiming at the problem of difficulty in selecting the proposal distribution and low computational efficiency in the conventional MCMC method, this paper proposed a simple population Metropolis–Hastings (SP-MH) algorithm for the damage identification, which is realized by exchanging information among chains in a relatively small population and using tuning-free strategy. Then, a numerical cantilever beam and an experimental frame are utilized to verify the effectiveness and feasibility of the proposed algorithm, it can be seen that the convergence rate of the SP-MH algorithm is faster than that of the Differential Evolution Monte Carlo (DE-MC) algorithm, and in a small population state, the SP-MH algorithm can still maintain convergence, saving plenty of computing time for damage identification. The results show that the SP-MH algorithm is feasible and accurate in practice damage identification, and the SP-MH algorithm performs better than the DE-MC algorithm. Compared with the DE-MC algorithm, the SP-MH algorithm is simple and convenient for damage identification due to its tuning-free strategy and relatively small population.

Probabilistic inversion for compressional modulus and shear modulus based on QA-MCMC algorithm with joint probability distribution

Compressional modulus and shear modulus play an important role in the reservoir prediction and fluid identification. Based on an optimized Markov chain Monte Carlo (MCMC) method in the Bayesian framework, we propose a probabilistic approach of utilizing seismic aptitudes to estimate compressional modulus and shear modulus. Conventional MCMC method is not efficient for prestack inversion due to the complex model parameter space. In order to accelerating the running speed of MCMC and improving the stability of Markov chain, we have developed a novel algorithm called “Quantum Annealing- Markov chain Monte Carlo algorithm (QA-MCMC) with joint probability distribution”. We derive an improved acceptance probability expression to reduce unnecessary model update in burn-in phase, and utilize joint probability distribution (JPD) to reduce the sampling space of inverted properties. Following a Bayesian framework, we generate the posterior probability distribution function and employ the improved MCMC algorithm to solve the inversion problem to estimate compressional modulus and shear modulus from prestack or angle stack gathers. The method is validated through the tests on the 1D model, 2D model and real data.

Bayesian Inversion of Logging-While-Drilling Extra-Deep Directional Resistivity Measurements Using Parallel Tempering Markov Chain Monte Carlo Sampling

We present a Bayesian inversion scheme to extract multiple bed boundaries from extra-deep directional logging-while-drilling (LWD) resistivity measurements (EDDRM). The inversion uses a dimensionality reduction approach to simplify the three-dimensional inversion problem into a stitch of 1-D ones. The Bayesian framework associated with a parallel tempering (PT) Markov Chain Monte Carlo (MCMC) sampling algorithm is invoked to derive the multiple boundaries from the 1-D inverse model. The kernel of PT algorithm is that multiple chains with different temperatures execute parallelly and the states can be swapped between the chains. Compared with conventional single-chain MCMC, the PT strategy accelerates the convergence and has a better global search capability. In addition, a new 1-D inverse model is proposed. By letting all beds share the same anisotropy coefficient, the new model incorporates fewer parameters of interests. Therefore, the MCMC samples can be reduced significantly. Numerical experiments performed over synthetic examples are presented to verify the feasibility of the new model, to test the inversion performance and to obtain the best practice of the inversion. The uncertainty of inverted results is also assessed from the probability distributions of resistivity profile and histograms of relative dipping and formation anisotropy.

An adaptive Kriging surrogate method for efficient joint estimation of hydraulic and biochemical parameters in reactive transport modeling

Groundwater reactive transport models that consider the coupling of hydraulic and biochemical processes are vital tools for predicting the fate of groundwater contaminants and effective groundwater management. The models involve a large number of parameters whose specification greatly affects the model performance. Thus model parameters calibration is crucial to its successful application. The Bayesian inference framework implemented by Markov chain Monte Carlo (MCMC) sampling provides a comprehensive framework to estimate the model parameters. However, its application is hampered by the large computational requirements caused by repeated evaluations of the model in MCMC sampling. This study develops an adaptive Kriging-based MCMC method to overcome the bottleneck of Bayesian inference by replacing the simulation model with a computationally inexpensive Kriging surrogate model. In the adaptive Kriging-based MCMC method, instead of constructing a globally accurate surrogate of the simulation model, we sequentially build a locally accurate surrogate with an iterative refinement to the high probability regions. The performance of the proposed method is demonstrated using a synthetic groundwater reactive transport model for describing sequential Kinetic degradation of Tetrachloroethene (PCE), whose hydraulic and biochemical parameters are jointly estimated. The results suggest that the adaptive Kriging-based MCMC method is able to achieve an accurate Bayesian inference with a hundredfold reduction in the computational cost compared to the conventional MCMC method.

Model Uncertainty Analysis for Metabolic Network Inference: A Case Study in Bayesian Model Averaging

A distinguishing feature of systems biology is the interrogation of models as a means of making predictions or generating deeper understanding of the systems under study. However, when using a given data set to address a specific question, a unique and provably correct model formulation to apply is rarely known. Instead, a large selection of alternative formulations of varying scopes ensues from combinatorial composition of entities. In this scenario, computational methods that allow us to make statistically valid inferences and predictions, while accounting for the uncertainty in model formulation are desired.We investigate into Bayesian Model Averaging (BMA), which accounts for model uncertainty by considering an ensemble of candidate models instead of a single model instance. To show the computational tractability of BMA, we perform model uncertainty analysis for a realistically sized reaction network from the domain of metabolic flux analysis, featuring an ensemble of millions of models. This is made possible using a Markov Chain Monte Carlo (MCMC) method, tailored to handle parameter and model structure uncertainty simultaneously. To investigate the computational burden of solving the multi-model problem, a super-model is created that includes the reactions of all models in the multi-model problem. The computational burden of the multi-model problem is compared to that of conventional MCMC inference on the single super-model. The comparison yields the surprising insight that the multi-model problem is computationally less expensive than the single super-model problem. Furthermore, we demonstrate, with the example at hand, that BMA yields valid structural network inferences.

Application of Hybrid Monte Carlo Algorithm in Heat Transfer

This article presents a new method of estimation of thermophysical parameters using the hybrid Monte Carlo (HMC) algorithm that synergistically combines the advantages of a Markov chain Monte Carlo (MCMC) method and molecular dynamics. The advantages of this technique over the conventional MCMC are elucidated by considering the multiparameter estimation in heat transfer. Four situations were analyzed. The first two involve a two- and a three-parameters estimation in a lumped capacitance model, third involves estimation in a distributed system, and the fourth involves estimation in a fin system. The goal is to establish the potency and usefulness of the HMC method for a wide class of engineering problems.

Zoeppritz-based AVO inversion using an improved Markov chain Monte Carlo method

The conventional Markov chain Monte Carlo (MCMC) method is limited to the selected shape and size of proposal distribution and is not easy to start when the initial proposal distribution is far away from the target distribution. To overcome these drawbacks of the conventional MCMC method, two useful improvements in MCMC method, adaptive Metropolis (AM) algorithm and delayed rejection (DR) algorithm, are attempted to be combined. The AM algorithm aims at adapting the proposal distribution by using the generated estimators, and the DR algorithm aims at enhancing the efficiency of the improved MCMC method. Based on the improved MCMC method, a Bayesian amplitude versus offset (AVO) inversion method on the basis of the exact Zoeppritz equation has been developed, with which the P- and S-wave velocities and the density can be obtained directly, and the uncertainty of AVO inversion results has been estimated as well. The study based on the logging data and the seismic data demonstrates the feasibility and robustness of the method and shows that all three parameters are well retrieved. So the exact Zoeppritz-based nonlinear inversion method by using the improved MCMC is not only suitable for reservoirs with strong-contrast interfaces and long-offset ranges but also it is more stable, accurate and anti-noise.

Automatically tuned general-purpose MCMC via new adaptive diagnostics

Adaptive Markov Chain Monte Carlo (MCMC) algorithms attempt to `learn' from the results of past iterations so the Markov chain can converge quicker. Unfortunately, adaptive MCMC algorithms are no longer Markovian, so their convergence is difficult to guarantee. In this paper, we develop new diagnostics to determine whether the adaption is still improving the convergence. We present an algorithm which automatically stops adapting once it determines further adaption will not increase the convergence speed. Our algorithm allows the computer to tune a `good' Markov chain through multiple phases of adaption, and then run conventional non-adaptive MCMC. In this way, the efficiency gains of adaptive MCMC can be obtained while still ensuring convergence to the target distribution.

Stochastic approximation Monte Carlo EM for change-point analysis

ABSTRACTIn the expectation–maximization (EM) algorithm for maximum likelihood estimation from incomplete data, Markov chain Monte Carlo (MCMC) methods have been used in change-point inference for a long time when the expectation step is intractable. However, the conventional MCMC algorithms tend to get trapped in local mode in simulating from the posterior distribution of change points. To overcome this problem, in this paper we propose a stochastic approximation Monte Carlo version of EM (SAMCEM), which is a combination of adaptive Markov chain Monte Carlo and EM utilizing a maximum likelihood method. SAMCEM is compared with the stochastic approximation version of EM and reversible jump Markov chain Monte Carlo version of EM on simulated and real datasets. The numerical results indicate that SAMCEM can outperform among the three methods by producing much more accurate parameter estimates and the ability to achieve change-point positions and estimates simultaneously.

Inference of regulatory networks with a convergence improved MCMC sampler

BackgroundOne of the goals of the Systems Biology community is to have a detailed map of all biological interactions in an organism. One small yet important step in this direction is the creation of biological networks from post-genomic data. Bayesian networks are a very promising model for the inference of regulatory networks in Systems Biology. Usually, Bayesian networks are sampled with a Markov Chain Monte Carlo (MCMC) sampler in the structure space. Unfortunately, conventional MCMC sampling schemes are often slow in mixing and convergence. To improve MCMC convergence, an alternative method is proposed and tested with different sets of data. Moreover, the proposed method is compared with the traditional MCMC sampling scheme.ResultsIn the proposed method, a simpler and faster method for the inference of regulatory networks, Graphical Gaussian Models (GGMs), is integrated into the Bayesian network inference, trough a Hierarchical Bayesian model. In this manner, information about the structure obtained from the data with GGMs is taken into account in the MCMC scheme, thus improving mixing and convergence. The proposed method is tested with three types of data, two from simulated models and one from real data. The results are compared with the results of the traditional MCMC sampling scheme in terms of network recovery accuracy and convergence. The results show that when compared with a traditional MCMC scheme, the proposed method presents improved convergence leading to better network reconstruction with less MCMC iterations.ConclusionsThe proposed method is a viable alternative to improve mixing and convergence of traditional MCMC schemes. It allows the use of Bayesian networks with an MCMC sampler with less iterations. The proposed method has always converged earlier than the traditional MCMC scheme. We observe an improvement in accuracy of the recovered networks for the Gaussian simulated data, but this improvement is absent for both real data and data simulated from ODE.Electronic supplementary materialThe online version of this article (doi:10.1186/s12859-015-0734-6) contains supplementary material, which is available to authorized users.

Seismic fluid identification using a nonlinear elastic impedance inversion method based on a fast Markov chain Monte Carlo method

Elastic impedance inversion with high efficiency and high stability has become one of the main directions of seismic pre-stack inversion. The nonlinear elastic impedance inversion method based on a fast Markov chain Monte Carlo (MCMC) method is proposed in this paper, combining conventional MCMC method based on global optimization with a preconditioned conjugate gradient (PCG) algorithm based on local optimization, so this method does not depend strongly on the initial model. It converges to the global optimum quickly and efficiently on the condition that efficiency and stability of inversion are both taken into consideration at the same time. The test data verify the feasibility and robustness of the method, and based on this method, we extract the effective pore-fluid bulk modulus, which is applied to reservoir fluid identification and detection, and consequently, a better result has been achieved.

An Overview of Stochastic Approximation Monte Carlo

During the past few decades, Markov chain Monte Carlo (MCMC) has been widely used in Bayesian statistical inference and scientific computing. Its successes have proven it to be a very powerful and typically unique computational tool for analyzing data of complex structures. However, conventional MCMC algorithms often suffer from the local trap problem which renders the simulation ineffective. This paper provides an overview of the theory, variants, and applications for stochastic approximation Monte Carlo (SAMC), an advanced MCMC algorithm that is essentially immune to the local trap problem. WIREs Comput Stat 2014, 6:240–254. doi: 10.1002/wics.1305This article is categorized under: Statistical and Graphical Methods of Data Analysis > Markov Chain Monte Carlo (MCMC)

An irreversible Markov-chain Monte Carlo method with skew detailed balance conditions

An irreversible Markov-chain Monte Carlo (MCMC) method based on a skew detailed balance condition is discussed. Some recent theoretical works concerned with the irreversible MCMC method are reviewed and the irreversible Metropolis-Hastings algorithm for the method is described. We apply the method to ferromagnetic Ising models in two and three dimensions. Relaxation dynamics of the order parameter and the dynamical exponent are studied in comparison to those with the conventional reversible MCMC method with the detailed balance condition. We also examine how the efficiency of exchange Monte Carlo method is affected by the combined use of the irreversible MCMC method.

An adaptive sparse-grid high-order stochastic collocation method for Bayesian inference in groundwater reactive transport modeling

[1] Bayesian analysis has become vital to uncertainty quantification in groundwater modeling, but its application has been hindered by the computational cost associated with numerous model executions required by exploring the posterior probability density function (PPDF) of model parameters. This is particularly the case when the PPDF is estimated using Markov Chain Monte Carlo (MCMC) sampling. In this study, a new approach is developed to improve the computational efficiency of Bayesian inference by constructing a surrogate of the PPDF, using an adaptive sparse-grid high-order stochastic collocation (aSG-hSC) method. Unlike previous works using first-order hierarchical basis, this paper utilizes a compactly supported higher-order hierarchical basis to construct the surrogate system, resulting in a significant reduction in the number of required model executions. In addition, using the hierarchical surplus as an error indicator allows locally adaptive refinement of sparse grids in the parameter space, which further improves computational efficiency. To efficiently build the surrogate system for the PPDF with multiple significant modes, optimization techniques are used to identify the modes, for which high-probability regions are defined and components of the aSG-hSC approximation are constructed. After the surrogate is determined, the PPDF can be evaluated by sampling the surrogate system directly without model execution, resulting in improved efficiency of the surrogate-based MCMC compared with conventional MCMC. The developed method is evaluated using two synthetic groundwater reactive transport models. The first example involves coupled linear reactions and demonstrates the accuracy of our high-order hierarchical basis approach in approximating high-dimensional posteriori distribution. The second example is highly nonlinear because of the reactions of uranium surface complexation, and demonstrates how the iterative aSG-hSC method is able to capture multimodal and non-Gaussian features of PPDF caused by model nonlinearity. Both experiments show that aSG-hSC is an effective and efficient tool for Bayesian inference.

Bayesian Parameter Inference by Markov Chain Monte Carlo with Hybrid Fitness Measures: Theory and Test in Apoptosis Signal Transduction Network

When exact values of model parameters in systems biology are not available from experiments, they need to be inferred so that the resulting simulation reproduces the experimentally known phenomena. For the purpose, Bayesian statistics with Markov chain Monte Carlo (MCMC) is a useful method. Biological experiments are often performed with cell population, and the results are represented by histograms. On another front, experiments sometimes indicate the existence of a specific bifurcation pattern. In this study, to deal with both type of such experimental results and information for parameter inference, we introduced functions to evaluate fitness to both type of experimental results, named quantitative and qualitative fitness measures respectively. We formulated Bayesian formula for those hybrid fitness measures (HFM), and implemented it to MCMC (MCMC-HFM). We tested MCMC-HFM first for a kinetic toy model with a positive feedback. Inferring kinetic parameters mainly related to the positive feedback, we found that MCMC-HFM reliably infer them with both qualitative and quantitative fitness measures. Then, we applied the MCMC-HFM to an apoptosis signal transduction network previously proposed. For kinetic parameters related to implicit positive feedbacks, which are important for bistability and irreversibility of the output, the MCMC-HFM reliably inferred these kinetic parameters. In particular, some kinetic parameters that have the experimental estimates were inferred without these data and the results were consistent with the experiments. Moreover, for some parameters, the mixed use of quantitative and qualitative fitness measures narrowed down the acceptable range of parameters. Taken together, our approach could reliably infer the kinetic parameters of the target systems.

A method for efficiently sampling from distributions with correlated dimensions.

Bayesian estimation has played a pivotal role in the understanding of individual differences. However, for many models in psychology, Bayesian estimation of model parameters can be difficult. One reason for this difficulty is that conventional sampling algorithms, such as Markov chain Monte Carlo (MCMC), can be inefficient and impractical when little is known about the target distribution--particularly the target distribution's covariance structure. In this article, we highlight some reasons for this inefficiency and advocate the use of a population MCMC algorithm, called differential evolution Markov chain Monte Carlo (DE-MCMC), as a means of efficient proposal generation. We demonstrate in a simulation study that the performance of the DE-MCMC algorithm is unaffected by the correlation of the target distribution, whereas conventional MCMC performs substantially worse as the correlation increases. We then show that the DE-MCMC algorithm can be used to efficiently fit a hierarchical version of the linear ballistic accumulator model to response time data, which has proven to be a difficult task when conventional MCMC is used.

변동진폭하중 하에서 균열성장예지를 위한 베이지안 모델변수 추정법

본 연구에서는 측정된 균열 데이터를 토대로 변동하중 하에서의 균열성장모델 변수들을 베이지안 모델변수 추정 방법을 통해서 확률적인 분포로 구하는 방법을 제시하였다. 모델변수의 확률분포를 구하기 위해 Markov Chain Monte Carlo (MCMC) 샘플링 방법을 이용하였다. 변동하중 하에서는 균열성장 모델이 더욱 복잡해 짐에 따라 기존의 MCMC 기법으로는 확률분포를 잘 구하지 못하므로 주변확률밀도분포를 제안함수로 사용하는 MCMC 기법을 새롭게 제안하였다. 모델변수의 추정을 위해 여러 크기의 일정 진폭 하중 하에서 시편시험을 수행하여 얻은 균열성장 데이터를 이용하였다. 추정된 변수들을 사용하여 변동하중 하에서의 시편에 대해 균열성장 예측을 수행하였고, 이를 실제 시험 데이터를 통해서 검증하였다.<BR>