Adaptive Metropolis-Hastings Research Articles

An investigation into Markov chain Monte Carlo algorithms for Subset simulation: Emphasizing uncertainty analysis

Subset simulation (SuS) has emerged as a crucial method for estimating failure probability, striking an efficient balance between accuracy and computational cost. This paper introduces an uncertainty analysis method for the Markov chain Monte Carlo (MCMC) algorithm within the SuS framework by examining three key aspects: Effective Sample Size (ESS) ratio, intra-chain correlation, and inter-chain correlation during SuS iterations. The first two affect the standard deviation and bias of the results when MCMC algorithms are run robustly, respectively, while the latter indicates whether the sample distribution generated from MCMC algorithms is stationary or not. Based on these insights, three MCMC algorithms are discussed: Component-wise adaptive Metropolis-Hastings (CW-aMH), Aadaptive Conditional sampling (aCS), and Elliptical Slice sampling (ES). It is demonstrated that better control of the low decreasing rate of ESS ratio in aCS results in a smaller standard deviation compared to CW-aMH in estimation. On the other hand, ES offers better unbiasedness and robustness owing to its low intra-chain correlation and inter-chain correlation. This method provides new insights into MCMC uncertainty analysis, establishing a potential statistical tool for future research in this area.

Gaussian process metamodel and Markov chain Monte Carlo-based Bayesian inference framework for stochastic nonlinear model updating with uncertainties

The estimation of the posterior probability density function (PDF) of unknown parameters remains a challenge in stochastic nonlinear model updating with uncertainties; thus, a novel Bayesian inference framework based on the Gaussian process metamodel (GPM) and the advanced Markov chain Monte Carlo (MCMC) method is proposed in this paper. The instantaneous characteristics (ICs) of the decomposed measurement response, calculated using the Hilbert transform and the discrete analytical mode decomposition methodology, are extracted as nonlinear indices and further used to construct the likelihood function. Then, the posterior PDFs of structural nonlinear model parameters are derived based on the Bayesian theorem. To precisely calculate the posterior PDF, an advanced MCMC approach, i.e., delayed rejection adaptive Metropolis-Hastings (DRAM) algorithm, is adopted with the advantages of a high acceptance ratio and strong robustness. However, as a common shortage in most MCMC methods, the resampling technology is still applied, and numerous iterations of nonlinear simulations are conducted to ensure accuracy, thus directly reducing the computational efficiency of the DRAM. To address the abovementioned issue, a mathematical regression metamodel of the GPM with a polynomial kernel function is adopted in this paper instead of the traditional finite element model (FEM) to simulate a nonlinear response for the reduction of computational cost, and the hyperparameters are further estimated using the conjugate gradient optimization methodology. Finally, numerical simulations concerning a Giuffré–Menegotto–Pinto modeled steel-frame structure and a seven-storey base-isolated structure are conducted. Furthermore, a shake-table experimental test of a nonlinear steel framework is investigated to validate the accuracy of the Bayesian inference method. Both simulations and experiment demonstrate that the proposed GPM and DRAM-based Bayesian method effectively estimates the posterior PDF of unknown parameters and is appropriate for stochastic nonlinear model updating even with multisource uncertainties.

Monte Carlo samplers for efficient network inference.

Accessing information on an underlying network driving a biological process often involves interrupting the process and collecting snapshot data. When snapshot data are stochastic, the data's structure necessitates a probabilistic description to infer underlying reaction networks. As an example, we may imagine wanting to learn gene state networks from the type of data collected in single molecule RNA fluorescence in situ hybridization (RNA-FISH). In the networks we consider, nodes represent network states, and edges represent biochemical reaction rates linking states. Simultaneously estimating the number of nodes and constituent parameters from snapshot data remains a challenging task in part on account of data uncertainty and timescale separations between kinetic parameters mediating the network. While parametric Bayesian methods learn parameters given a network structure (with known node numbers) with rigorously propagated measurement uncertainty, learning the number of nodes and parameters with potentially large timescale separations remain open questions. Here, we propose a Bayesian nonparametric framework and describe a hybrid Bayesian Markov Chain Monte Carlo (MCMC) sampler directly addressing these challenges. In particular, in our hybrid method, Hamiltonian Monte Carlo (HMC) leverages local posterior geometries in inference to explore the parameter space; Adaptive Metropolis Hastings (AMH) learns correlations between plausible parameter sets to efficiently propose probable models; and Parallel Tempering takes into account multiple models simultaneously with tempered information content to augment sampling efficiency. We apply our method to synthetic data mimicking single molecule RNA-FISH, a popular snapshot method in probing transcriptional networks to illustrate the identified challenges inherent to learning dynamical models from these snapshots and how our method addresses them.

Generalized fiducial inference for the lower confidence limit of reliability based on Weibull distribution

Reliability performance, especially the lower confidence limit of reliability, plays an important role in reliability assessment, and it is also of concern to researchers and engineers. In this article, a novel estimate of the lower confidence limit of the reliability for two-parameter Weibull distribution is proposed based on the generalized fiducial inference. The corresponding adaptive Metropolis-Hastings within Gibbs algorithm is provided to analyze the generalized fiducial lower confidence limit. The proposed lower confidence limit of the reliability is compared with that from frequentist method. In addition, this novel procedure is generalized to the series system and the parallel system which consists of identical Weibull components. Simulation results show that the proposed generalized fiducial lower confidence limit is more applicable than the frequentist method, especially for the case of small sample size. Finally, the MOS transistor lifetime data is used to illustrate the new procedure.

Model Updating of a Prestressed Concrete Rigid Frame Bridge Using Multiple Markov Chain Monte Carlo Method and Differential Evolution

Model updating is a widely adopted method to minimize the error between test results from the real structure and outcomes from the finite element (FE) model for obtaining an accurate and reliable FE model of the target structure. However, uncertainties from the environment, excitation and measurement variability can reduce the accuracy of predictions of the updated FE model. The Bayesian model updating method using multiple Markov chains based on differential evolution adaptive metropolis (DREAM) algorithm is explored, which runs multiple chains simultaneously for a global exploration, and it automatically tunes the scale and orientation of the proposal distribution during the evolution of the posterior distribution. The performance of the proposed method is illustrated numerically with a beam model and a three-span rigid frame bridge. Results show that the DREAM algorithm is capable for updating the FE model in civil engineering. It extends the Bayesian model updating method to multiple Markov chains scenario, which provides higher accuracy than single chain algorithm such as the delayed rejection adaptive metropolis-hastings (DRAM) method. Moreover, results from both examples indicate that the proposed method is insensitive to values of initial parameters, which avoid errors resulting from inappropriate prior knowledge of parameters in the FE model updating.

A novel Bayesian functional spatial partitioning method with application to prostate cancer lesion detection using MRI.

Spatial partitioning methods correct for nonstationarity in spatially related data by partitioning the space into regions of local stationarity. Existing spatial partitioning methods can only estimate linear partitioning boundaries. This is inadequate for detecting an arbitrarily shaped anomalous spatial region within a larger area. We propose a novel Bayesian functional spatial partitioning (BFSP) algorithm, which estimates closed curves that act as partitioning boundaries around anomalous regions of data with a distinct distribution or spatial process. Our method utilizes transitions between a fixed Cartesian and moving polar coordinate system to model the smooth boundary curves using functional estimation tools. Using adaptive Metropolis-Hastings, the BFSP algorithm simultaneously estimates the partitioning boundary and the parameters of the spatial distributions within each region. Through simulation we show that our method is robust to shape of the target zone and region-specific spatial processes. We illustrate our method through the detection of prostate cancer lesions using magnetic resonanceimaging.

Markov Chain Monte Carlo-based Bayesian method for nonlinear stochastic model updating

This paper proposes a Markov Chain Monte Carlo (MCMC)-based Bayesian method for nonlinear stochastic model updating by using the instantaneous characteristics of the structural dynamic responses. According to the discrete analytical mode decomposed method and Hilbert transform, the instantaneous characteristics of the mono-components are firstly extracted from the structural dynamic response and applied to the calculation of likelihood function. Then, the posterior probability density function associated with Bayesian theorem is derived under the assumption of Gaussian prior distribution by using instantaneous characteristics. Afterwards, to calculate the posterior probability density function and improve the sampling efficiency, the delayed rejection adaptive Metropolis-Hastings (DRAM) algorithm is implemented with the advantages of strong adaptive and fast convergence. In the process of Bayesian inference, the posterior samples generated by DRAM require vast quantities of finite element analysis to guarantee the accuracy. For reducing the computational cost, the response surface model is constructed to establish the mathematical regression model between the structural parameters and the theoretical dynamic responses. To validate the effectiveness and applicability of the proposed method, the numerical cases on a three-story nonlinear structure under earthquake excitation considering various noise level effects and an Iwan beam model with two types of excitations are simulated. In addition, an experimental validation on a ¼ scale, 3-story steel frame structure subjected to a series of earthquake excitations in the laboratory is also performed to further verify the proposed method. Both the numerical and experimental results demonstrate that the DRAM-based Bayesian method can be effectively used to update nonlinear stochastic models with a high accuracy.

Bayesian approach to parameter estimation and model validation for nuclear fusion reactor mean-field edge turbulence modelling

This paper presents a Bayesian approach to infer about two mean-field plasma turbulence models, a first based on the turbulent kinetic energy k ⊥, and a second based on k ⊥ and the turbulent enstrophy ζ ⊥. These models contain several closure terms with unknown constants that have to be determined through fitting to reference data from turbulence simulations or experiments. In this paper, we compare two techniques to solve the Bayesian inference problem: the Laplace approximation and the adaptive Metropolis–Hastings (AMH) algorithm. Our Bayesian inference allows for parameter uncertainty quantification, identification of parameter cross-correlations and model comparison through the Bayesian evidence. Our results indicate that while a diffusive k ⊥–ζ ⊥ scaling for the anomalous diffusion coefficient provides a better approximation to the turbulent particle flux when based on exact turbulence simulation data, at present large modelling uncertainties and parameter cross-correlations in the full k ⊥–ζ ⊥ model make it less performant than the more simple k ⊥ model. For the cases studied here, the cross-correlations can be removed by a reparameterization of the k ⊥–ζ ⊥ model with fewer parameters. The results can form the basis for further development of the turbulence models.

Parameter optimization and uncertainty assessment for rainfall frequency modeling using an adaptive Metropolis-Hastings algorithm.

A new parameter optimization and uncertainty assessment procedure using the Bayesian inference with an adaptive Metropolis-Hastings (AM-H) algorithm is presented for extreme rainfall frequency modeling. An efficient Markov chain Monte Carlo sampler is adopted to explore the posterior distribution of parameters and calculate their uncertainty intervals associated with the magnitude of estimated rainfall depth quantiles. Also, the efficiency of AM-H and conventional maximum likelihood estimation (MLE) in parameter estimation and uncertainty quantification are compared. And the procedure was implemented and discussed for the case of Chaohu city, China. Results of our work reveal that: (i) the adaptive Bayesian method, especially for return level associated to large return period, shows better estimated effect when compared with MLE; it should be noted that the implementation of MLE often produces overy optimistic results in the case of Chaohu city; (ii) AM-H algorithm is more reliable than MLE in terms of uncertainty quantification, and yields relatively narrow credible intervals for the quantile estimates to be instrumental in risk assessment of urban storm drainage planning.

Colombian Women’s Life Patterns: A Multivariate Density Regression Approach

Women in Colombia face difficulties related to the patriarchal traits of their societies and well-known conflict afflicting the country since 1948. In this critical context, our aim is to study the relationship between baseline socio-demographic factors and variables associated to fertility, partnership patterns, and work activity. To best exploit the explanatory structure, we propose a Bayesian multivariate density regression model, which can accommodate mixed responses with censored, constrained, and binary traits. The flexible nature of the models allows for nonlinear regression functions and non-standard features in the errors, such as asymmetry or multi-modality. The model has interpretable covariate-dependent weights constructed through normalization, allowing for combinations of categorical and continuous covariates. Computational difficulties for inference are overcome through an adaptive truncation algorithm combining adaptive Metropolis-Hastings and sequential Monte Carlo to create a sequence of automatically truncated posterior mixtures. For our study on Colombian women’s life patterns, a variety of quantities are visualised and described, and in particular, our findings highlight the detrimental impact of family violence on women’s choices and behaviors.

Scalable Bayesian inference for self-excitatory stochastic processes applied to big American gunfire data.

The Hawkes process and its extensions effectively model self-excitatory phenomena including earthquakes, viral pandemics, financial transactions, neural spike trains and the spread of memes through social networks. The usefulness of these stochastic process models within a host of economic sectors and scientific disciplines is undercut by the processes' computational burden: complexity of likelihood evaluations grows quadratically in the number of observations for both the temporal and spatiotemporal Hawkes processes. We show that, with care, one may parallelize these calculations using both central and graphics processing unit implementations to achieve over 100-fold speedups over single-core processing. Using a simple adaptive Metropolis-Hastings scheme, we apply our high-performance computing framework to a Bayesian analysis of big gunshot data generated in Washington D.C. between the years of 2006 and 2019, thereby extending a past analysis of the same data from under 10,000 to over 85,000 observations. To encourage widespread use, we provide hpHawkes, an open-source R package, and discuss high-level implementation and program design for leveraging aspects of computational hardware that become necessary in a big data setting.

Geoacoustic inversion of the acoustic-pressure vertical phase gradient from a single vector sensor.

A vector sensor can provide measurements of ocean acoustic fields in terms of the acoustic pressure and three-dimensional particle velocity, providing potentially highly-informative data for applications such as geoacoustic inversion. This paper applies nonlinear Bayesian inversion to vector sensor data to estimate seabed geoacoustic properties and uncertainties in South China Sea. Linear-frequency-modulated source transmissions, recorded as acoustic pressure and vertical particle velocity, are processed to estimate the vertical phase gradient of acoustic pressure at multiple frequencies as the inversion data. An advantage of this type of data is that it can be modeled without knowledge of the source spectrum, allowing inversion with an unknown source and a single sensor. Geoacoustic inversion of phase-gradient data is carried out and compared to inversion of the vertical acoustic impedance, another type of vector-sensor data, independent of the source spectrum, which has been considered previously. Model selection for the optimal number of seabed sediment layers is carried out using Bayesian information criterion, and parameter estimates, uncertainties, and correlations are calculated using delayed-rejection adaptive Metropolis-Hastings sampling. Results indicate a three-layer seabed model (including the semi-infinite basement), with properties in agreement with independent measurements including a high-resolution seismic profile and surficial sediment type from a core.

An Adaptive Metropolis-Hastings Optimization Algorithm of Bayesian Estimation in Non-Stationary Flood Frequency Analysis

Global climate changing and human activities have altered the assumption of stationarity, and intensified the variation of hydrological process in recent decades. It is essential to make progress in accommodating appropriate models to the changing environment where non-stationary models are taken into account. The developing adapted Bayesian inference offers an attractive framework to estimate non-stationary models, when compared with conventional maximum likelihood estimation (MLE). As the inseparable companions of Bayesian inference, an efficient MCMC sampler are expected to be built. However, proper tunings are needed for the sampler to improve the performance by integrating adaptive algorithm and optimization method. A Bayesian approach with the adaptive Metropolis-Hastings optimization (AM-HO) algorithm is adopted to estimate the parameters and quantify the uncertainty in a two-parameter non-stationary Lognormal distribution model. To verify the performance of the developed model, simulation experiments and practical applications are implemented to fit annual maximum flood series of two gauges in Hanjiang River basin. From the point view of parameters estimation, both Bayesian and MLE methods perform similarly. However, Bayesian method is more attractive and reliable than MLE on uncertainty quantification, which provides a relative narrow intervals to be beneficial for risk analysis and water resource management.

Assessing uncertainties in velocity models and images with a fast nonlinear uncertainty quantification method

Seismic imaging is conventionally performed using noisy data and a presumably inexact velocity model. Uncertainties in the input parameters propagate directly into the final image and therefore into any quantity of interest, or qualitative interpretation, obtained from the image. We considered the problem of uncertainty quantification in velocity building and seismic imaging using Bayesian inference. Using a reduced velocity model, a fast field expansion method for simulating recorded wavefields, and the adaptive Metropolis-Hastings algorithm, we efficiently quantify velocity model uncertainty by generating multiple models consistent with low-frequency full-waveform data. A second application of Bayesian inversion to any seismic reflections present in the recorded data reconstructs the corresponding structures’ position along with its associated uncertainty. Our analysis complements rather than replaces traditional imaging because it allows us to assess the reliability of visible image features and to take that into account in subsequent interpretations.

Model Selection and Adaptive Markov Chain Monte Carlo for Bayesian Cointegrated VAR Model

In this paper, we develop novel Markov chain Monte Carlo sampling methodology for Bayesian Cointegrated Vector Auto Regression (CVAR) models. Here we focus on two novel exten sions to the sampling methodology for the CVAR posterior distribution. The first extension we develop replaces the popular sampling methodology of the griddy Gibbs sampler with an automated alternative which is based on an Adaptive Metropolis-Hastings algorithm. This is particularly relevant to automate the proposal mechanism in the MCMC algorithm in settings where griddy Gibbs is impractical such as when the dimension of the CVAR series is large, e.g. d > 5. We also treat the rank of the CVAR model as a random variable and perform joint inference on the rank and model parameters. This is achieved with a Bayesian posterior distribution defined over both the rank and the CVAR model parameters, and inference is made via a Savage-Dickey density estimator for the Bayes Factor analysis of rank.

Land leveling impact on surface runoff and soil losses: Estimation with coupled deterministic/stochastic models for a Québec agricultural field

Land leveling impact on water quality had not received much attention for fields in humid continental climate. The objectives of this study were to isolate the impact of land leveling, performed on an agricultural field (Québec, Canada) in spring 2012, on runoff and TSS load and to make recommendations to attenuate adverse environmental impacts of land leveling, if any. A total of 66 runoff events, including 22 with total suspended sediments (TSS) load estimates, from 2010 to 2014 were analyzed. To this end, deterministic models were coupled to an adaptive Metropolis-Hastings algorithm to estimate the unknown distribution of the parameters representing the most important effects, namely land leveling, tillage, and crop cover. Simulated runoff events were generated by the hydrological model SWMM version 5 while simulated TSS loads were generated by an empirical equation based on the Revised Universal Soil Loss Equation version 2 (RUSLE2). Thanks to the algorithm used, it was demonstrated that land leveling significantly decreased total runoff volume at least for the two following years. The impact on peak flow was mixed: land leveling significantly decreased peak flow for a typical stratiform rainfall event but the effect was unclear for a typical convective rainfall event. Based on 90% confidence interval, TSS load increased from 10 to 1000 times immediately after land leveling (spring 2012) compared to pre-land leveling events. The TSS load increase remained significant one year after land leveling, with TSS loads 5–20 times higher compared to pre-land leveling events. It would thus be recommended to grow crops with high ground coverage ratios coupled with cover crops during the year when land leveling is done. Sediment retention structures could also be installed at the beginning of the land leveling process to provide protection against the short term and delayed impact on water quality.

Fitting complex population models by combining particle filters with Markov chain Monte Carlo

We show how a recent framework combining Markov chain Monte Carlo (MCMC) with particle filters (PFMCMC) may be used to estimate population state-space models. With the purpose of utilizing the strengths of each method, PFMCMC explores hidden states by particle filters, while process and observation parameters are estimated using an MCMC algorithm. PFMCMC is exemplified by analyzing time series data on a red kangaroo (Macropus rufus) population in New South Wales, Australia, using MCMC over model parameters based on an adaptive Metropolis-Hastings algorithm. We fit three population models to these data; a density-dependent logistic diffusion model with environmental variance, an unregulated stochastic exponential growth model, and a random-walk model. Bayes factors and posterior model probabilities show that there is little support for density dependence and that the random-walk model is the most parsimonious model. The particle filter Metropolis-Hastings algorithm is a brute-force method that may be used to fit a range of complex population models. Implementation is straightforward and less involved than standard MCMC for many models, and marginal densities for model selection can be obtained with little additional effort. The cost is mainly computational, resulting in long running times that may be improved by parallelizing the algorithm.

Efficient Bayesian Inference for Multiple Change-Point and Mixture Innovation Models

Time series subject to parameter shifts of random magnitude and timing are commonly modeled with a change-point approach using Chib's (1998) algorithm to draw the break dates. We outline some advantages of an alternative approach in which breaks come through mixture distributions in state innovations, and for which the sampler of Gerlach, Carter and Kohn (2000) allows reliable and efficient inference. We show how the same sampler can be used to (i) model shifts in variance that occur independently of shifts in other parameters (ii) draw the break dates in O(n) rather than O(n³) operations in the change-point model of Koop and Potter (2004b), the most general to date. Finally, we introduce to the time series literature the concept of adaptive Metropolis-Hastings sampling for discrete latent variable models. We develop an easily implemented adaptive algorithm that improves on Gerlach et al. (2000) and promises to significantly reduce computing time in a variety of problems including mixture innovation, change-point, regime-switching, and outlier detection. The efficiency gains on two models for U.S. inflation and real interest rates are 257% and 341%.