Traditional Markov Chain Monte Carlo Research Articles

Comparison of Machine-learning and Bayesian Inferences for the Interior of Rocky Exoplanets with Large Compositional Diversity

In previous work, we demonstrated that machine-learning techniques based on mixture density networks (MDNs) are successful in inferring the interior structure of rocky exoplanets with large compositional diversity. In this study, we compare the performance of a well-trained MDN model with the conventional Bayesian inversion method based on the Markov chain Monte Carlo (MCMC) method, under the same observable constraints. Considering that MCMC inversion is generally performed with the prior knowledge of planetary mass, radius, and bulk molar ratios of Fe/Mg and Si/Mg, we regenerate a substantial data set of interior structure data for rocky exoplanets and train a new MDN model with inputs of planetary mass, radius, Fe/Mg, and Si/Mg. It has been found that the well-trained MDN model has comparable performance to that of the MCMC method but requires significantly less computation time. The MDN model presents a practical alternative to the traditional MCMC method, surpassing the latter with minimal requirements for specialized knowledge, faster prediction, and greater adaptability. The developed MDN model is made publicly available on GitHub for the broader scientific community’s utilization. With the advent of the James Webb Space Telescope, we are ushering in a new epoch in exoplanetary explorations. In this evolving landscape, the MDN model stands out as a valuable asset, particularly for its ability to rapidly assimilate and interpret new data, thereby substantially advancing our understanding of the interior and habitability of exoplanetary systems.

Spatial Modeling of Air Pollution Using Data Fusion

Air pollution is a widespread issue. One approach to predicting air pollution levels in specific locations is through the development of mathematical models. Spatial models are one such category, and they can be optimized using calculation methods like the INLA (integrated nested Laplace approximation) package. It streamlines the complex computational process by combining the Laplace approximation and numerical integration to approximate the model and provides a computationally efficient alternative to traditional MCMC (Markov chain Monte Carlo) methods for Bayesian inference in complex hierarchical models. Another crucial aspect is obtaining data for this type of problem. Relying only on official or professional monitoring stations can pose challenges, so it is advisable to employ data fusion techniques and integrate data from various sensors, including amateur ones. Moreover, when modeling spatial air pollution, careful consideration should be given to factors such as the range of impact and potential obstacles that may affect a pollutant’s dispersion. This study showcases the utilization of INLA spatial modeling and data fusion to address multiple problems, such as pollution in industrial facilities and urban areas. The results show promise for resolving such problems with the proposed algorithms.

APES: approximate posterior ensemble sampler

ABSTRACT This paper proposes a novel approach to generate samples from target distributions that are difficult to sample from using Markov chain Monte Carlo (MCMC) methods. Traditional MCMC algorithms often face slow convergence due to the difficulty in finding proposals that suit the problem at hand. To address this issue, the paper introduces the approximate posterior ensemble sampler (APES) algorithm, which employs kernel density estimation and radial basis interpolation to create an adaptive proposal, leading to fast convergence of the chains. The APES algorithm’s scalability to higher dimensions makes it a practical solution for complex problems. The proposed method generates an approximate posterior probability that closely approximates the desired distribution and is easy to sample from, resulting in smaller autocorrelation times and a higher probability of acceptance by the chain. We compare the performance of the APES algorithm with the affine invariance ensemble sampler with the stretch move in various contexts, demonstrating the efficiency of the proposed method. For instance, on the Rosenbrock function, the APES presented an autocorrelation time 140 times smaller than the affine invariance ensemble sampler. The comparison showcases the effectiveness of the APES algorithm in generating samples from challenging distributions. This paper presents a practical solution to generating samples from complex distributions while addressing the challenge of finding suitable proposals. With new cosmological surveys set to deal with many new systematics, this method offers a practical solution for the upcoming era of cosmological analyses. The algorithms presented in this paper are available at https://github.com/NumCosmo/NumCosmo.

Sampling by divergence minimization

We introduce a Markov Chain Monte Carlo (MCMC) method that is designed to sample from target distributions with irregular geometry using an adaptive scheme. In cases where targets exhibit non-Gaussian behavior, we propose that adaptation should be regional rather than global. Our algorithm minimizes the information projection component of the Kullback–Leibler (KL) divergence between the proposal and target distributions to encourage proposals that are distributed similarly to the regional geometry of the target. Unlike traditional adaptive MCMC, this procedure rapidly adapts to the geometry of the target’s current position as it explores the surrounding space without the need for many preexisting samples. The divergence minimization algorithms are tested on target distributions with irregularly shaped modes and we provide results demonstrating the effectiveness of our methods.

Research on an Optimized Evaluation Method of the Bearing Capacity of Reinforced Concrete Beam Based on the Bayesian Theory

An optimized evaluation method of the bearing capacity of reinforced concrete beam based on the Bayesian theory was proposed in this paper. This evaluation method optimized the traditional Markov Chain-Monte Carlo (MCMC) sampling method, and proposed an improved Metropolis-Hastings (MH) sampling method and a transitive MCMC (TMCMC) sampling method based on the MCMC theory. These two derived sampling methods solved the problem that the traditional MCMC algorithm makes it difficult to achieve convergence when the number of modified parameters is large. Therefore, on the basis of obtaining the measured sample information and the prior information of uncertain parameters, this paper first used multiple "model components" to form a model sample, then carried out a sensitivity analysis based on the relevant response indicators and selected the key parameters that had a great impact on the bearing capacity, carried out static load tests, and extracted and analyzed the experimental data. Then, based on a large amount of analysis data, the improved MH sampling method and TMCMC sampling method were used to establish a posterior probability distribution database. Finally, multiple posterior probability distributions were used to identify and predict the bearing capacity. The results showed that the method was feasible and effective for the evaluation of the bearing capacity of reinforced concrete beam.

Extremely expensive likelihoods: a variational-Bayes solution for precision cosmology

ABSTRACTWe present a variational-Bayes solution to compute non-Gaussian posteriors from extremely expensive likelihoods. Our approach is an alternative for parameter inference when Markov chain Monte Carlo (MCMC) sampling is numerically prohibitive or conceptually unfeasible. For example, when either the likelihood or the theoretical model cannot be evaluated at arbitrary parameter values, but only previously selected values, then traditional MCMC sampling is impossible, whereas our variational-Bayes solution still succeeds in estimating the full posterior. In cosmology, this occurs e.g. when the parametric model is based on costly simulations that were run for previously selected input parameters. We demonstrate the applicability of our posterior construction on the KiDS-450 weak lensing analysis, where we reconstruct the original KiDS MCMC posterior at 0.6 per cent of its former numerical posterior evaluations. The reduction in numerical cost implies that systematic effects which formerly exhausted the numerical budget could now be included.

A multi-scale blocking moving window algorithm for geostatistical seismic inversion

Markov chain Monte Carlo (McMC) methods are suitable for solving the high-dimensional geostatistical seismic inverse problem. However, traditional McMC is impractical to generate desired lithofacies distribution since the repeated geostatistics and seismic forward simulators are very time-consuming. Recently, various strategies (e.g., the sequential geostatistical resampling (SGR) idea, multi-scale strategy, and annealing process) have been individually designed to alleviate the computational burden. However, the coordination between multiple corresponding key parameters (e.g., blocking window sizes, grid level, and temperatures) adds difficulties to combining multiple strategies with McMC. In this context, we propose a Multi-scale Blocking Moving Window algorithm (MsBMW) to effectively combines blocking McMC updating, a multi-scale strategy, and a new simulated annealing (SA) algorithm. In the iterative process, we monitor the mean acceptance rate (AR) to update window sizes, grid level, and temperatures, keeping the AR within the desired range (e.g., between 25% and 50%) for more efficient sampling. To assess the performance of the MsBMW, we tested it on the Standford VI synthetic reservoir. The standard Metropolis-Hastings (MH) sampling is performed to present the posterior probability density function (pdf) as the reference in this study. Compared with the Blocking Moving Window algorithm (BMW) proposed by Alcolea and Renard (2010), the MsBMW allows better quality results in less time. The results show that the multi-scale approach has the effect of accelerating the convergence of inversion. The adaptive cooling schedule circumvents the problem that the cooling rate is difficult to control in SA. Finally, we applied the proposed methodology to two-dimensional (2-D) and three-dimensional (3-D) real study areas and quickly generated multiple realizations that fit geological patterns, well data, and seismic data for uncertainty evaluation.

Approximate Methods for Bayesian Computation

Rich data generating mechanisms are ubiquitous in this age of information and require complex statistical models to draw meaningful inference. While Bayesian analysis has seen enormous development in the last 30 years, benefitting from the impetus given by the successful application of Markov chain Monte Carlo (MCMC) sampling, the combination of big data and complex models conspire to produce significant challenges for the traditional MCMC algorithms. We review modern algorithmic developments addressing the latter and compare their performance using numerical experiments.

A Hybrid Approach for Approximating the Ideal Observer for Joint Signal Detection and Estimation Tasks by Use of Supervised Learning and Markov-Chain Monte Carlo Methods.

The ideal observer (IO) sets an upper performance limit among all observers and has been advocated for assessing and optimizing imaging systems. For general joint detection and estimation (detection-estimation) tasks, estimation ROC (EROC) analysis has been established for evaluating the performance of observers. However, in general, it is difficult to accurately approximate the IO that maximizes the area under the EROC curve. In this study, a hybrid method that employs machine learning is proposed to accomplish this. Specifically, a hybrid approach is developed that combines a multi-task convolutional neural network and a Markov-Chain Monte Carlo (MCMC) method in order to approximate the IO for detection-estimation tasks. Unlike traditional MCMC methods, the hybrid method is not limited to use of specific utility functions. In addition, a purely supervised learning-based sub-ideal observer is proposed. Computer-simulation studies are conducted to validate the proposed method, which include signal-known-statistically/background-known-exactly and signal-known-statistically/background-known-statistically tasks. The EROC curves produced by the proposed method are compared to those produced by the MCMC approach or analytical computation when feasible. The proposed method provides a new approach for approximating the IO and may advance the application of EROC analysis for optimizing imaging systems.

An Efficient Coalescent Epoch Model for Bayesian Phylogenetic Inference.

We present a two-headed approach called Bayesian Integrated Coalescent Epoch PlotS (BICEPS) for efficient inference of coalescent epoch models. Firstly, we integrate out population size parameters, and secondly, we introduce a set of more powerful Markov chain Monte Carlo (MCMC) proposals for flexing and stretching trees. Even though population sizes are integrated out and not explicitly sampled through MCMC, we are still able to generate samples from the population size posteriors. This allows demographic reconstruction through time and estimating the timing and magnitude of population bottlenecks and full population histories. Altogether, BICEPS can be considered a more muscular version of the popular Bayesian skyline model. We demonstrate its power and correctness by a well-calibrated simulation study. Furthermore, we demonstrate with an application to SARS-CoV-2 genomic data that some analyses that have trouble converging with the traditional Bayesian skyline prior and standard MCMC proposals can do well with the BICEPS approach. BICEPS is available as open-source package for BEAST 2 under GPL license and has a user-friendly graphical user interface.[Bayesian phylogenetics; BEAST 2; BICEPS; coalescent model.].

Scaling Bayesian inference of mixed multinomial logit models to large datasets

Variational inference methods have been shown to lead to significant improvements in the computational efficiency of approximate Bayesian inference in mixed multinomial logit models when compared to standard Markov-chain Monte Carlo (MCMC) methods without increasing estimation bias. However, despite their demonstrated efficiency gains, existing methods still suffer from important limitations that prevent them to scale to large datasets, while providing the flexibility to allow for rich prior distributions and to capture complex posterior distributions. To effectively scale Bayesian inference in Mixed Multinomial Logit models to large datasets, we propose an Amortized Variational Inference approach that leverages stochastic backpropagation, automatic differentiation and GPU-accelerated computation. Moreover, we show how normalizing flows can be used to increase the flexibility of the variational posterior approximations. Through an extensive simulation study and real data for transport mode choice from London, we empirically show that the proposed approach is able to achieve computational speedups of multiple orders of magnitude over traditional maximum simulated likelihood estimation (MSLE) and MCMC approaches for large datasets without compromising estimation accuracy.

A reduced order approach for probabilistic inversions of 3-D magnetotelluric data I: general formulation

SUMMARYSimulation-based probabilistic inversions of 3-D magnetotelluric (MT) data are arguably the best option to deal with the nonlinearity and non-uniqueness of the MT problem. However, the computational cost associated with the modelling of 3-D MT data has so far precluded the community from adopting and/or pursuing full probabilistic inversions of large MT data sets. In this contribution, we present a novel and general inversion framework, driven by Markov Chain Monte Carlo (MCMC) algorithms, which combines (i) an efficient parallel-in-parallel structure to solve the 3-D forward problem, (ii) a reduced order technique to create fast and accurate surrogate models of the forward problem and (iii) adaptive strategies for both the MCMC algorithm and the surrogate model. In particular, and contrary to traditional implementations, the adaptation of the surrogate is integrated into the MCMC inversion. This circumvents the need of costly offline stages to build the surrogate and further increases the overall efficiency of the method. We demonstrate the feasibility and performance of our approach to invert for large-scale conductivity structures with two numerical examples using different parametrizations and dimensionalities. In both cases, we report staggering gains in computational efficiency compared to traditional MCMC implementations. Our method finally removes the main bottleneck of probabilistic inversions of 3-D MT data and opens up new opportunities for both stand-alone MT inversions and multi-observable joint inversions for the physical state of the Earth’s interior.

Facilitating Bayesian analysis of combustion kinetic models with artificial neural network

Bayesian analysis provides a framework for the inverse uncertainty quantification (UQ) of combustion kinetic models. As the workhorse of the Bayesian approach, the Markov chain Monte Carlo (MCMC) methods, however, incur a substantial computational cost. In this work, a surrogate model is employed to improve the traditional MCMC algorithm. Specifically, the test errors of three typical surrogate models are compared, namely Polynomial Chaos Expansion (PCE), High Dimensional Model Representation (HDMR) and Artificial Neural Network (ANN); and ANN is shown to be a relatively more efficient surrogate model for the approximation of combustion reaction systems under extensive conditions. An inverse UQ method, which is the combination of the ANN and traditional MCMC method, and as such termed ANN–MCMC, is adopted. The calculation is performed on the methanol oxidation system and a series of ignition delay data are employed to optimize the rate coefficients of the kinetic model. The estimated posterior distributions of the rate coefficients and the model predictions using the ANN–MCMC are compared with the traditional MCMC methods, with the results showing that the ANN–MCMC can significantly reduce the computational cost needed for the MCMC algorithm, especially on the estimation of the posterior distributions of the input parameters. The rejection rate of the samples in a Markov chain is very high, especially for the calculation of the posterior distribution of less sensitive parameters, thus a large number of samples are needed to reach a desired accuracy for traditional MCMC process. While no samples are rejected when training the ANN surrogate model. Therefore, fewer original samples are needed to get a converged ANN surrogate, which can then generate a large number of low-cost ANN samples for a better accuracy of the MCMC process. The errors for the estimated posterior distributions using ANN–MCMC depend on the accuracy of converged ANN surrogates and more accurate results are obtained with improved settings of ANN. The ANN–MCMC is especially suitable to the computational systems when the computational ability is limited.

Probabilistic power flow based on slice sampling for distribution network containing distributed generations

With the wide application of distributed generations (DG) in power system, some problems appear to the power flow calculation of distribution network. In the methods of probabilistic power flow calculation based on Monte Carlo simulation (MCS), the Gibbs sampling algorithm needs a large number of complex iterative operations to get more accurate results. Aiming at the problem of the algorithm, a Markov Chain Monte Carlo (MCMC) simulation method based on slice sampling algorithm is proposed and applied to probabilistic power flow calculation of the distribution network containing distributed generation. Finally, the IEEE-33 node system is used for simulation. The results show that the slice sampling algorithm can significantly improve the computational accuracy of the traditional MCMC method. In the meantime, the slice sampling is faster and more stable than Gibbs sampling under the same number of sampling iterations.

Markov chain Monte Carlo ground‐motion selection algorithms for conditional intensity measure targets

SummaryTwo new algorithms are presented for efficiently selecting suites of ground motions that match a target multivariate distribution or conditional intensity measure target. The first algorithm is a Markov chain Monte Carlo (MCMC) approach in which records are sequentially added to a selected set such that the joint probability density function (PDF) of the target distribution is progressively approximated by the discrete distribution of the selected records. The second algorithm derives from the concept of the acceptance ratio within MCMC but does not involve any sampling. The first method takes advantage of MCMC's ability to efficiently explore a sampling distribution through the implementation of a traditional MCMC algorithm. This method is shown to enable very good matches to multivariate targets to be obtained when the numbers of records to be selected is relatively large. A weaker performance for fewer records can be circumvented by the second method that uses greedy optimisation to impose additional constraints upon properties of the target distribution. A preselection approach based upon values of the multivariate PDF is proposed that enables near‐optimal record sets to be identified with a very close match to the target. Both methods are applied for a number response analyses associated with different sizes of record sets and rupture scenarios. Comparisons are made throughout with the Generalised Conditional Intensity Measure (GCIM) approach. The first method provides similar results to GCIM but with slightly worse performance for small record sets, while the second method outperforms method 1 and GCIM for all considered cases.

Approximate Bayesian computation for spatial SEIR(S) epidemic models

Approximate Bayesia n Computation (ABC) provides an attractive approach to estimation in complex Bayesian inferential problems for which evaluation of the kernel of the posterior distribution is impossible or computationally expensive. These highly parallelizable techniques have been successfully applied to many fields, particularly in cases where more traditional approaches such as Markov chain Monte Carlo (MCMC) are impractical. In this work, we demonstrate the application of approximate Bayesian inference to spatially heterogeneous Susceptible-Exposed-Infectious-Removed (SEIR) stochastic epidemic models. These models have a tractable posterior distribution, however MCMC techniques nevertheless become computationally infeasible for moderately sized problems. We discuss the practical implementation of these techniques via the open source ABSEIR package for R. The performance of ABC relative to traditional MCMC methods in a small problem is explored under simulation, as well as in the spatially heterogeneous context of the 2014 epidemic of Chikungunya in the Americas.

A new methodology for Bayesian history matching using parallel interacting Markov chain Monte Carlo

This paper presents an innovative application of a new class of parallel interacting Markov chains Monte Carlo to solve the Bayesian history matching (BHM) problem. BHM consists of sampling a posterior distribution given by the Bayesian theorem. Markov chain Monte Carlo (MCMC) is well suited for sampling, in principle, any type of distribution; however the number of iteration required by the traditional single-chain MCMC can be prohibitive in BHM applications. Furthermore, history matching is typically a highly nonlinear inverse problem, which leads in very complex posterior distributions, characterized by many separated modes. Therefore, single chain can be trapped into a local mode. Parallel interacting chains is an interesting way to overcome this problem, as shown in this paper. In addition, we presented new approaches to define starting points for the parallel chains. For validation purposes, the proposed methodology is firstly applied in a simple but challenging cross section reservoir model with many modes in the posterior distribution. Afterwards, the application to a realistic case integrated to geostatistical modelling is also presented. The results showed that the combination of parallel interacting chain with the capabilities of distributed computing commonly available nowadays is very promising to solve the BHM problem.

Bayesian System Identification using auxiliary stochastic dynamical systems

Bayesian approaches to statistical inference and system identification became practical with the development of effective sampling methods like Markov Chain Monte Carlo (MCMC). However, because the size and complexity of inference problems has dramatically increased, improved MCMC methods are required. Dynamical systems based samplers are an effective extension of traditional MCMC methods. These samplers treat the posterior probability distribution as the potential energy function of a dynamical system, enabling them to better exploit the structure of the inference problem. We present an algorithm, Second-Order Langevin MCMC (SOL-MC), a stochastic dynamical system based MCMC algorithm, which uses the damped second-order Langevin stochastic differential equation (SDE) to sample a posterior distribution. We design the SDE such that the desired posterior probability distribution is its stationary distribution. Since this method is based upon an underlying dynamical system, we can utilize existing work to develop, implement, and optimize the sampler's performance. As such, we can choose parameters which speed up the convergence to the stationary distribution and reduce temporal state and energy correlations in the samples. We then apply this sampler to a system identification problem for a non-linear hysteretic structure model to investigate this method under globally identifiable and unidentifiable conditions.

A Two-Tier Wind Power Time Series Model Considering Day-to-Day Weather Transition and Intraday Wind Power Fluctuations

A novel two-tier wind power time series model considering day-to-day weather transition and intraday wind power fluctuations is proposed. Weather factors and conditions are classified into typical weather states in terms of the effects on wind power using a fuzzy clustering technique. A typical weather Markov chain model is established to characterize day-to-day weather transition process. An improved Markov Chain Monte Carlo (MCMC) model considering the probability distributions of the wind power at the first time point of each day and wind power fluctuations is developed to characterize intraday wind power fluctuation process. The day-to-day typical weather Markov chain and intraday wind power time series for each typical weather state are simulated separately first and then integrated into a complete wind power time series. The proposed model is verified using the wind power records and weather data at an actual wind farm. The results indicate that different weather states have significantly different impacts on the distributions of daily average wind powers. The comparison analysis confirms that although the additional weather data inputs and the increase of model parameters are requisite, the proposed model outperforms the ARIMA model and the traditional MCMC model in terms of various accuracy indices.

Source identification of sudden contamination based on the parameter uncertainty analysis

It is important to identify the source information after a sudden water contamination incident occurs in a water supply system. The accuracy of the simulation model's parameters determines the accuracy of the source information. However, it is difficult to obtain the true value of these parameters by existing methods, so reduction of the errors caused by the uncertainty of these parameters is a crucial problem. A source identification framework which considers the uncertainty of the model's sensitive parameters and combines Bayesian inference and Markov Chain Monte Carlo (MCMC) algorithms simulation is established, and the South-to-North Water Diversion Project is taken as the case study in this paper. Compared with a framework which does not consider the uncertainty of the model's parameters, the proposed framework could solve the error caused by the wrong choice of model parameters and obtain more accurate results. In addition, the proposed framework based on traditional MCMC and that based on the Delayed Rejection and Adaptive Metropolis (DRAM-MCMC) are compared to prove that the DRAM-MCMC is more convergent and accurate. Lastly, the proposed framework based on DRAM-MCMC is proved to solve the problem with high practicality and generality in the studied long distance water diversion project.