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Bayesian methods have been widely used in the last two decades to infer statistical properties of spatially variable coefficients in partial differential equations from measurements of the solutions of these equations. Yet, in many cases the number of variables used to parameterize these coefficients is large, and oobtaining meaningful statistics of their probability distributions is difficult using simple sampling methods such as the basic Metropolis- Hastings algorithm--in particular, if the inverse problem is ill-conditioned or ill-posed. As a consequence, many advanced sampling methods have been described in the literature that converge faster than Metropolis-Hastings, for example, by exploiting hierarchies of statistical models or hierarchies of discretizations of the underlying differential equation. At the same time, it remains difficult for the reader of the literature to quantify the advantages of these algorithms because there is no commonly used benchmark. This paper presents a benchmark Bayesian inverse problem--namely, the determination of a spatially variable coefficient, discretized by 64 values, in a Poisson equation, based on point measurements of the solution--that fills the gap between widely used simple test cases (such as superpositions of Gaussians) and real applications that are difficult to replicate for developers of sampling algorithms. We provide a complete description of the test case and provide an open-source implementation that can serve as the basis for further experiments. We have also computed 2 x 1011 samples, at a cost of some 30 CPU years, of the posterior probability distribution from which we have generated detailed and accurate statistics against which other sampling algorithms can be tested. [ABSTRACT FROM AUTHOR]

This article develops a Bayesian computational platform at the interface between posterior sampling and optimization in models whose marginal likelihoods are difficult to evaluate. Inspired by contrastive learning and Generative Adversarial Networks (GAN), we reframe the likelihood function estimation problem as a classification problem. Pitting a Generator, who simulates fake data, against a Classifier, who tries to distinguish them from the real data, one obtains likelihood (ratio) estimators which can be plugged into the Metropolis–Hastings algorithm. The resulting Markov chains generate, at a steady state, samples from an approximate posterior whose asymptotic properties we characterize. Drawing upon connections with empirical Bayes and Bayesian misspecification, we quantify the convergence rate in terms of the contraction speed of the actual posterior and the convergence rate of the Classifier. Asymptotic normality results are also provided which justify the inferential potential of our approach. We illustrate the usefulness of our approach on examples which have challenged for existing Bayesian likelihood-free approaches. for this article are available online. [ABSTRACT FROM AUTHOR]

We develop a new Markov chain on graph partitions that makes relatively global moves yet is computationally feasible to be used as the proposal in the Metropolis--Hastings method. Our resulting algorithm is able to sample from a specified measure on partitions or spanning forests. Being able to sample from a specified measure is a requirement of what we consider as the gold standard in quantifying the extent to which a particular map is a gerrymander. Our proposal chain modifies the recently developed method called recombination (ReCom), which draws spanning trees on joined partitions and then randomly cuts them to repartition. We improve the computational efficiency by augmenting the statespace from partitions to spanning forests. The extra information accelerates the computation of the forward and backward proposal probabilities which are required for the Metropolis--Hastings algorithm. We demonstrate this method by sampling redistricting plans on several measures of interest and find promising convergence results on several key observables of interest. We also explore some limitations in the measures that are efficient to sample from and investigate the feasibility of using parallel tempering to extend this space of measures. [ABSTRACT FROM AUTHOR]

Various Markov chain Monte Carlo (MCMC) methods are studied to improve upon random walk Metropolis sampling, for simulation from complex distributions. Examples include Metropolis-adjusted Langevin algorithms, Hamiltonian Monte Carlo, and other algorithms related to underdamped Langevin dynamics. We propose a broad class of irreversible sampling algorithms, called Hamiltonian-assisted Metropolis sampling (HAMS), and develop two specific algorithms with appropriate tuning and preconditioning strategies. Our HAMS algorithms are designed to simultaneously achieve two distinctive properties, while using an augmented target density with a momentum as an auxiliary variable. One is generalized detailed balance, which induces an irreversible exploration of the target. The other is a rejection-free property for a Gaussian target with a prespecified variance matrix. This property allows our preconditioned algorithms to perform satisfactorily with relatively large step sizes. Furthermore, we formulate a framework of generalized Metropolis–Hastings sampling, which not only highlights our construction of HAMS at a more abstract level, but also facilitates possible further development of irreversible MCMC algorithms. We present several numerical experiments, where the proposed algorithms consistently yield superior results among existing algorithms using the same preconditioning schemes. [ABSTRACT FROM AUTHOR]

The problem of optimally scaling the proposal distribution in a Markov chain Monte Carlo algorithm is critical to the quality of the generated samples. Much work has gone into obtaining such results for various Metropolis–Hastings (MH) algorithms. Recently, acceptance probabilities other than MH are being employed in problems with intractable target distributions. There are few resources available on tuning the Gaussian proposal distributions for this situation. We obtain optimal scaling results for a general class of acceptance functions, which includes Barker's and lazy MH. In particular, optimal values for Barker's algorithm are derived and found to be significantly different from that obtained for the MH algorithm. Our theoretical conclusions are supported by numerical simulations indicating that when the optimal proposal variance is unknown, tuning to the optimal acceptance probability remains an effective strategy. [ABSTRACT FROM AUTHOR]

Markov Chain Monte Carlo simulations form an essential tool for exploring high‐dimensional target distributions. Metropolis developed a fundamental random walk algorithm which was improved by Hastings later. The result is known as the Metropolis‐Hastings algorithm, which enables the exploration of multi‐dimensional distributions. The main drawbacks of this algorithm are its high auto‐correlation and slow exploration of the target distribution space. In order to increase efficiency, researchers have proposed various modifications to this algorithm. In particular, the Hamiltonian Monte Carlo simulation enhances the efficient exploration of the target probability density. The algorithm uses mechanisms inspired by Hamiltonian dynamics to propose a new sample for the target distribution. For reliability analysis, the incorporation of subset simulation and Hamiltonian Monte Carlo methods has shown promising results. However, using the Hamiltonian Monte Carlo method to sample is computationally expensive, especially when dealing with high‐dimensional problems and performing several steps to propose a new state. In this contribution, we show the general applicability of Hamiltonian neural networks to speed up the proposal of new samples within the Hamiltonian Monte Carlo method. [ABSTRACT FROM AUTHOR]

Methods of estimating statistical distribution have attracted many researchers when it comes to fitting a specific distribution to data. However, when the data belong to more than one component, a popular distribution cannot be fitted to such data. To tackle this issue, mixture models are fitted by choosing the correct number of components that represent the data. This can be obvious in lifetime processes that are involved in a wide range of engineering applications as well as biological systems. In this paper, we introduce an application of estimating a finite mixture of Inverse Rayleigh distribution by the use of the Bayesian framework when considering the model as Markov chain Monte Carlo (MCMC). We employed the Gibbs sampler and Metropolis – Hastings algorithms. The proposed techniques are applied to simulated data following several scenarios. The accuracy of estimation has been examined by the average mean square error (AMSE) and the average classification success rate (ACSR). The results showed that the method was well performed in all simulation scenarios with respect to different sample sizes. [ABSTRACT FROM AUTHOR]

Repulsive mixture models have recently gained popularity for Bayesian cluster detection. Compared to more traditional mixture models, repulsive mixture models produce a smaller number of well-separated clusters. The most commonly used methods for posterior inference either require to fix a priori the number of components or are based on reversible jump MCMC computation. We present a general framework for mixture models, when the prior of the "cluster centers" is a finite repulsive point process depending on a hyperparameter, specified by a density which may depend on an intractable normalizing constant. By investigating the posterior characterization of this class of mixture models, we derive a MCMC algorithm which avoids the well-known difficulties associated to reversible jump MCMC computation. In particular, we use an ancillary variable method, which eliminates the problem of having intractable normalizing constants in the Hastings ratio. The ancillary variable method relies on a perfect simulation algorithm, and we demonstrate this is fast because the number of components is typically small. In several simulation studies and an application on sociological data, we illustrate the advantage of our new methodology over existing methods, and we compare the use of a determinantal or a repulsive Gibbs point process prior model. Supplementary files for this article are available online. [ABSTRACT FROM AUTHOR]

Traditional Markov chain Monte Carlo (MCMC) algorithms are computationally intensive and do not scale well to large data. In particular, the Metropolis–Hastings (MH) algorithm requires passing over the entire dataset to evaluate the likelihood ratio in each iteration. We propose a general framework for performing MH-MCMC using mini-batches of the whole dataset and show that this gives rise to approximately a tempered stationary distribution. We prove that the algorithm preserves the modes of the original target distribution and derive an error bound on the approximation with mild assumptions on the likelihood. To further extend the utility of the algorithm to high-dimensional settings, we construct a proposal with forward and reverse moves using stochastic gradient and show that the construction leads to reasonable acceptance probabilities. We demonstrate the performance of our algorithm in both low dimensional models and high dimensional neural network applications. Particularly in the latter case, compared to popular optimization methods, our method is more robust to the choice of learning rate and improves testing accuracy. for this article are available online. [ABSTRACT FROM AUTHOR]

We develop a Metropolized Multiscale Forest Recombination Markov Chain on redistricting plans. The chain is designed to be usable as the proposal in a Markov Chain Monte Carlo (MCMC) algorithm. Sampling the space of plans amounts to dividing a graph into a partition with a specified number of elements each of which corresponds to a different district according to a specified probability measure. The districts satisfy a collection of hard constraints, and the probability measure may be weighted with regard to a number of other criteria. The multiscale algorithm is similar to our previously developed Metropolized Forest Recombination proposal; however, this algorithm provides improved scaling properties and may also be used to preserve nested communities of interest such as counties and precincts. Both works use a proposal which extends the ReCom algorithm [D. DeFord, M. Duchin, and J. Solomon, Harvard Data Sci. Rev., (2021)] which leveraged spanning trees to merge and split districts. In this work, we extend the state space so that each district is defined by a hierarchy of trees. In this sense, the proposal step in both algorithms can be seen as a "Forest ReCom." The collection of plans sampled by the MCMC algorithm can serve as a baseline against which a particular plan of interest is compared. If a given plan has different racial or partisan qualities than what is typical of the collection of plans, the given plan may have been gerrymandered and is labeled as an outlier. Metropolizing relative to a policy driven probability measure removes the possibility of algorithmically inserted biases. [ABSTRACT FROM AUTHOR]

The purpose of this article is to find a better technique for estimating the volatility of the price of bitcoin on the one hand and to check if this special kind of asset called cryptocurrency behaves like other stock market indices. We include five stock market indexes for different countries such as Standard and Poor's 500 composite Index (S&P), Nasdaq, Nikkei, Stoxx, and DowJones. Using daily data over the period 2010–2019. We examine two asymmetric stochastic volatility models used to describe the volatility dependencies found in most financial returns. Two models are compared, the first is the autoregressive stochastic volatility model with Student's t-distribution (ARSV-t), and the second is the basic SVOL. To estimate these models, our analysis is based on the Markov Chain Monte-Carlo method. Therefore, the technique used is a Metropolis–Hastings (Hastings in Biometrika 57:97–109, 1970), and the Gibbs sampler (Casella and George in Am Stat 46:167–174, 1992; Gelfand and Smith in J Am Stat Assoc 85:398–409, 1990; Gilks and Wild in 41:337–348, 1992). Model comparisons illustrate that the ARSV-t model performs better performances. We conclude that this model is better than the SVOL model on the MSE and AIC function. This result concerns bitcoin as well as the other stock market indices. Without forgetting that our finding proves the efficiency of Markov Chain for our sample and the convergence and stability for all parameters to a certain level. On the whole, it seems that permanent shocks have an effect on the volatility of the price of the bitcoin and also on the other stock market. Our results will help investors better diversify their portfolio by adding this cryptocurrency. [ABSTRACT FROM AUTHOR]

Bayesian inference in the presence of an intractable likelihood function is computationally challenging. When following a Markov chain Monte Carlo (MCMC) approach to approximate the posterior distribution in this context, one typically either uses MCMC schemes which target the joint posterior of the parameters and some auxiliary latent variables, or pseudo-marginal Metropolis--Hastings (MH) schemes. The latter mimic a MH algorithm targeting the marginal posterior of the parameters by approximating unbiasedly the intractable likelihood. However, in scenarios where the parameters and auxiliary variables are strongly correlated under the posterior and/or this posterior is multimodal, Gibbs sampling or Hamiltonian Monte Carlo (HMC) will perform poorly and the pseudo-marginal MH algorithm, as any other MH scheme, will be inefficient for high-dimensional parameters. We propose here an original MCMC algorithm, termed pseudo-marginal HMC, which combines the advantages of both HMC and pseudo-marginal schemes. Specifically, the PM-HMC method is controlled by a precision parameter N, controlling the approximation of the likelihood and, for any N, it samples the marginal posterior of the parameters. Additionally, as N tends to infinity, its sample trajectories and acceptance probability converge to those of an ideal, but intractable, HMC algorithm which would have access to the intractable likelihood and its gradient. We demonstrate through experiments that PM-HMC can outperform significantly both standard HMC and pseudo-marginal MH schemes. [ABSTRACT FROM AUTHOR]

One main limitation of the existing optimal scaling results for Metropolis–Hastings algorithms is that the assumptions on the target distribution are unrealistic. In this paper, we consider optimal scaling of random-walk Metropolis algorithms on general target distributions in high dimensions arising from practical MCMC models from Bayesian statistics. For optimal scaling by maximizing expected squared jumping distance (ESJD), we show the asymptotically optimal acceptance rate 0.234 can be obtained under general realistic sufficient conditions on the target distribution. The new sufficient conditions are easy to be verified and may hold for some general classes of MCMC models arising from Bayesian statistics applications, which substantially generalize the product i.i.d. condition required in most existing literature of optimal scaling. Furthermore, we show one-dimensional diffusion limits can be obtained under slightly stronger conditions, which still allow dependent coordinates of the target distribution. We also connect the new diffusion limit results to complexity bounds of Metropolis algorithms in high dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we derive a Markov chain Monte Carlo (MCMC) algorithm supported by a neural network. In particular, we use the neural network to substitute derivative calculations made during a Metropolis adjusted Langevin algorithm (MALA) step with inexpensive neural network evaluations. Using a complex, high-dimensional blood coagulation model and a set of measurements, we define a likelihood function on which we evaluate the new MCMC algorithm. The blood coagulation model is a dynamic model, where derivative calculations are expensive and hence limit the efficiency of derivative-based MCMC algorithms. The MALA adaptation greatly reduces the time per iteration, while only slightly affecting the sample quality. We also test the new algorithm on a 2-dimensional example with a non-convex shape, a case where the MALA algorithm has a clear advantage over other state of the art MCMC algorithms. To assess the impact of the new algorithm, we compare the results to previously generated results of the MALA and the random walk Metropolis Hastings (RWMH). [ABSTRACT FROM AUTHOR]

We propose a fast potential splitting Markov chain Monte Carlo method which costs O(1) time each step for sampling from equilibrium distributions (Gibbs measures) corresponding to particle systems with singular interacting kernels. We decompose the interacting potential into two parts; one is of long range but is smooth, and the other one is of short range but may be singular. To displace a particle, we first evolve a selected particle using the stochastic differential equation (SDE) under the smooth part with the idea of random batches, as commonly used in stochastic gradient Langevin dynamics. Then, we use the short range part to do a Metropolis rejection. Different from the classical Langevin dynamics, we only run the SDE dynamics with a random batch for a short duration of time so that the cost in the first step is O(p), where p is the batch size and is often chosen to be O(1). The cost of the rejection step is O(1) since the interaction used is of short range. We justify the proposed random-batch Monte Carlo method, which combines the random batch and splitting strategies, both in theory and with numerical experiments. While giving comparable results for typical examples of the Dyson Brownian motion and Lennard-Jones fluids, our method can save more time when compared to the classical Metropolis-Hastings algorithm. [ABSTRACT FROM AUTHOR]

Designing robust mutation strategies for primary sample space Metropolis light transport is a challenging problem: poorly tuned mutations both hinder state space exploration and introduce structured image artifacts. Scenes with complex materials, lighting, and geometry make hand-designing strategies that remain optimal over the entire state space infeasible. Moreover, these difficult regions are often sparse in state space, and so relying exclusively on intricate—and often expensive—proposal mechanisms can be wasteful, whereas simpler inexpensive mechanisms are more sample efficient. We generalize Metropolis–Hastings light transport to employ a flexible two-stage mutation strategy based on delayed rejection Markov chain Monte Carlo. Our approach generates multiple proposals based on the failure of previous ones, all while preserving Markov chain ergodicity. This allows us to reduce error while maintaining fast global exploration and low correlation across chains. Direct application of delayed rejection to light transport leads to low acceptance probabilities, and so we also propose a novel transition kernel to alleviate this issue. We benchmark our approach on several applications including bold-then-timid and cheap-then-expensive proposals across different light transport algorithms. Our method is applicable to any primary sample space algorithm with minimal implementation effort, producing consistently better results on a variety of challenging scenes. [ABSTRACT FROM AUTHOR]

Optimization-based samplers such as randomize-then-optimize (RTO) [J. M. Bardsley et al., SIAM J. Sci. Comput., 36 (2014), pp. A1895--A1910] provide an efficient and parallellizable approach to solving large-scale Bayesian inverse problems. These methods solve randomly perturbed optimization problems to draw samples from an approximate posterior distribution. ''Correcting"" these samples, either by Metropolization or importance sampling, enables characterization of the original posterior distribution. This paper focuses on the scalability of RTO to problems with highor infinite-dimensional parameters. In particular, we introduce a new subspace strategy to reformulate RTO. For problems with intrinsic low-rank structures, this subspace acceleration makes the computational complexity of RTO scale linearly with the parameter dimension. Furthermore, this subspace perspective suggests a natural extension of RTO to a function space setting. We thus formalize a function space version of RTO and establish sufficient conditions for it to produce a valid Metropolis--Hastings proposal, yielding dimension-independent sampling performance. Numerical examples corroborate the dimension independence of RTO and demonstrate sampling performance that is also robust to small observational noise. [ABSTRACT FROM AUTHOR]

In a 1970 Biometrika paper, W. K. Hastings developed a broad class of Markov chain algorithms for sampling from probability distributions that are difficult to sample from directly. The algorithm draws a candidate value from a proposal distribution and accepts the candidate with a probability that can be computed using only the unnormalized density of the target distribution, allowing one to sample from distributions known only up to a constant of proportionality. The stationary distribution of the corresponding Markov chain is the target distribution one is attempting to sample from. The Hastings algorithm generalizes the Metropolis algorithm to allow a much broader class of proposal distributions instead of just symmetric cases. An important class of applications for the Hastings algorithm corresponds to sampling from Bayesian posterior distributions, which have densities given by a prior density multiplied by a likelihood function and divided by a normalizing constant equal to the marginal likelihood. The marginal likelihood is typically intractable, presenting a fundamental barrier to implementation in Bayesian statistics. This barrier can be overcome by Markov chain Monte Carlo sampling algorithms. Amazingly, even after 50 years, the majority of algorithms used in practice today involve the Hastings algorithm. This article provides a brief celebration of the continuing impact of this ingenious algorithm on the 50th anniversary of its publication. [ABSTRACT FROM AUTHOR]

Markov Chain Monte Carlo (MCMC) simulation has a considerable computational burden when the target probability density function (PDF) evaluation involves a black-box, potentially computationally-expensive, numerical model. A novel framework to accelerate MCMC is developed here for such applications. It leverages a Kriging surrogate model approximation to the target PDF to improve computational efficiency, while preserves convergence properties to the exact target PDF, avoiding potential accuracy problems introduced through the surrogate model error. The approach relies on the delayed-rejection (DR) scheme and combines the rapid exploration characteristics of global (independent) proposals with the local search robustness of random walk proposals under the MCMC setting. The global proposal is chosen as the Kriging-based target PDF approximation. This proposal may resemble, depending on the surrogate model error, the actual target PDF very well, and therefore can provide significant computational benefits, including fast convergence of the Markov chain to its stationary distribution and, more importantly, low correlation between samples. At each MCMC step a candidate sample is generated from the independent proposal. If rejected, DR allows an extra random walk around the current sample. The DR step guarantees convergence to the actual target PDF, circumventing robustness problems that may arise when the Kriging-based independent proposal offers a poor approximation to (i.e., underestimates) the actual target PDF. The overall computational efficiency is further improved through an adaptive updating of the Kriging surrogate model during the MCMC sampling phase by extracting information from candidate samples whose inclusion in the training database can substantially enhance Kriging's accuracy. The computational efficiency and robustness of the established algorithm, termed Adaptive Kriging Delayed Rejection Adaptive Metropolis algorithm(AK-DRAM), are demonstrated in two analytical benchmark problems and two engineering problems focusing on conditional failure sample simulation and Bayesian inference. • An efficient implementation of MCMC is established using Metropolis–Hastings as sampling algorithm. • Foundation is a Kriging-based approximation of the actual target density. • An independent proposal density using this approximation facilitates efficient global exploration. • Delayer rejection through a local random walk guarantees convergence to actual target density. • Adaptive refinement of the Kriging-based density approximation further improves efficiency. [ABSTRACT FROM AUTHOR]

Coupling Bayes' Theorem with a two-dimensional (2D) groundwater solute advection-diffusion transport equation allows an inverse model to be established to identify a set of contamination source parameters including source intensity (M), release location (X0, Y0) and release time (T0), based on monitoring well data. To address the issues of insufficient monitoring wells or weak correlation between monitoring data and model parameters, a monitoring well design optimization approach was developed based on the Bayesian formula and information entropy. To demonstrate how the model works, an exemplar problem with an instantaneous release of a contaminant in a confined groundwater aquifer was employed. The information entropy of the model parameters posterior distribution was used as a criterion to evaluate the monitoring data quantity index. The optimal monitoring well position and monitoring frequency were solved by the two-step Monte Carlo method and differential evolution algorithm given a known well monitoring locations and monitoring events. Based on the optimized monitoring well position and sampling frequency, the contamination source was identified by an improved Metropolis algorithm using the Latin hypercube sampling approach. The case study results show that the following parameters were obtained: 1) the optimal monitoring well position (D) is at (445, 200); and 2) the optimal monitoring frequency (Δt) is 7, providing that the monitoring events is set as 5 times. Employing the optimized monitoring well position and frequency, the mean errors of inverse modeling results in source parameters (M, X0,Y0,T0) were 9.20%, 0.25%, 0.0061%, and 0.33%, respectively. The optimized monitoring well position and sampling frequency can effectively safeguard the inverse modeling results in identifying the contamination source parameters. It was also learnt that the improved Metropolis-Hastings algorithm (a Markov chain Monte Carlo method) can make the inverse modeling result independent of the initial sampling points and achieves an overall optimization, which significantly improved the accuracy and numerical stability of the inverse modeling results. [ABSTRACT FROM AUTHOR]

In this paper we address the problem of the prohibitively large computational cost of existing Markov chain Monte Carlo methods for large-scale applications with high-dimensional parameter spaces, e.g., in uncertainty quantification in porous media flow. We propose a new multilevel Metropolis--Hastings algorithm and give an abstract, problem-dependent theorem on the cost of the new multilevel estimator based on a set of simple, verifiable assumptions. For a typical model problem in subsurface flow, we then provide a detailed analysis of these assumptions and show significant gains over the standard Metropolis--Hastings estimator. Numerical experiments confirm the analysis and demonstrate the effectiveness of the method with consistent reductions of more than an order of magnitude in the cost of the multilevel estimator over the standard Metropolis--Hastings algorithm for tolerances ϵ < 10-2. [ABSTRACT FROM AUTHOR]

One of the most widely used samplers in practice is the component-wise Metropolis–Hastings (CMH) sampler that updates in turn the components of a vector-valued Markov chain using accept–reject moves generated from a proposal distribution. When the target distribution of a Markov chain is irregularly shaped, a "good" proposal distribution for one region of the state–space might be a "poor" one for another region. We consider a component-wise multiple-try Metropolis (CMTM) algorithm that chooses from a set of candidate moves sampled from different distributions. The computational efficiency is increased using an adaptation rule for the CMTM algorithm that dynamically builds a better set of proposal distributions as the Markov chain runs. The ergodicity of the adaptive chain is demonstrated theoretically. The performance is studied via simulations and real data examples. Supplementary material for this article is available online. [ABSTRACT FROM AUTHOR]

• A PCHM approach is developed for hydrological data assimilation. • A mixed evolution scheme is employed in PCMH to generate balanced samples. • PCMH can generate more reliable predictions with less accuracy variations. In this study, a particle copula Metropolis-Hastings (PCMH) approach was developed for reliable uncertainty quantification of hydrological predictions. The proposed PCMH approach employs a mixed particle evolution scheme, which integrates the Gaussian perturbation and copula-based dependent sampling methods. The Metropolis ratio is then employed to determine the acceptance of the candidate samples. The applicability of PCMH is elaborated for a long-term data assimilation case at the River Ouse in UK. Multiple hydrological models and different uncertainty settings in inputs, outputs and sample sizes are tested by the PCMH, particle filter (PF) and particle Markov chain Monte Carlo (PMCMC) approaches. The results indicate that the developed PCMH approach is able to generate more reliable results with less accuracy fluctuation than PF and PMCMC for both deterministic and probabilistic predictions from all the hydrological models. The mean values and the associated variation intervals of NSE over the total 270 runs for PCMH, PF and PMCMC are 0.752 (variation interval of [0.534, 0.866]), 0.661 (variation interval of [0.080, 0.879]), and 0.655 (variation interval of [0.247, 0.824]), respectively. For the probabilistic predictions evaluated by CRPS, the mean values and fluctuation ranges from PCMH, PF and PMCMC are respectively 15.215 ([8.624,31.549]), 18.758 ([8.595, 43.536]), 19.308 ([10.848, 37.799]). These results suggested that the proposed PCMH method would be more robust than PF and PMCMC in generating reliable hydrologic predictions and be less influenced by the hydrologic model structures, uncertainty scenarios, and its inherent randomness. Moreover, the PCMH method can also show better robustness than the copula-based particle filter method since the particle evolution scheme of PCMH would balance extreme samples from copula sampling procedure by mixing samples from Gaussian perturbation and remove unacceptable candidates through the Metropolis acceptance criterion. [ABSTRACT FROM AUTHOR]

Markov chain Monte Carlo (MCMC) techniques are usually used to infer model parameters when closed-form inference is not feasible, with one of the simplest MCMC methods being the random walk Metropolis–Hastings (MH) algorithm. The MH algorithm suffers from random walk behaviour, which results in inefficient exploration of the target posterior distribution. This method has been improved upon, with algorithms such as Metropolis Adjusted Langevin Monte Carlo (MALA) and Hamiltonian Monte Carlo being examples of popular modifications to MH. In this work, we revisit the MH algorithm to reduce the autocorrelations in the generated samples without adding significant computational time. We present the: (1) Stochastic Volatility Metropolis–Hastings (SVMH) algorithm, which is based on using a random scaling matrix in the MH algorithm, and (2) Locally Scaled Metropolis–Hastings (LSMH) algorithm, in which the scaled matrix depends on the local geometry of the target distribution. For both these algorithms, the proposal distribution is still Gaussian centred at the current state. The empirical results show that these minor additions to the MH algorithm significantly improve the effective sample rates and predictive performance over the vanilla MH method. The SVMH algorithm produces similar effective sample sizes to the LSMH method, with SVMH outperforming LSMH on an execution time normalised effective sample size basis. The performance of the proposed methods is also compared to the MALA and the current state-of-art method being the No-U-Turn sampler (NUTS). The analysis is performed using a simulation study based on Neal's funnel and multivariate Gaussian distributions and using real world data modeled using jump diffusion processes and Bayesian logistic regression. Although both MALA and NUTS outperform the proposed algorithms on an effective sample size basis, the SVMH algorithm has similar or better predictive performance when compared to MALA and NUTS across the various targets. In addition, the SVMH algorithm outperforms the other MCMC algorithms on a normalised effective sample size basis on the jump diffusion processes datasets. These results indicate the overall usefulness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Bayesian estimation of nonlinear stochastic differential equations when observations are discretely sampled. The estimation framework relies on the introduction of latent auxiliary data to complete the missing diffusion between each pair of measurements. Tuned Markov chain Monte Carlo (MCMC) methods based on the Metropolis‐Hastings algorithm, in conjunction with the Euler‐Maruyama discretization scheme, are used to sample the posterior distribution of the latent data and the model parameters. Techniques for computing the likelihood function, the marginal likelihood, and diagnostic measures (all based on the MCMC output) are developed. Examples using simulated and real data are presented and discussed in detail. [ABSTRACT FROM AUTHOR]

• Bayesian Probabilistic calibration of an Ni-Ti precipitation model by MCMC method. • Finding a function between interfacial energy and input variables by MCMC method. • Using cokriging surrogate modeling for a better prediction of precipitation behavior. Ni-Ti alloys are the most popular shape memory alloys in different industrial applications due to their especial properties provided, for at least some variants, by the precipitation of secondary strengthening phases. Therefore, precipitation modeling and associated model calibration coupled with uncertainty estimations of their performance seem to be crucial for these alloys, particularly when the focus is on the design of precipitation schemes leading to optimal performance. In this work, the calibration and uncertainty quantification of influential model parameters (i.e. the parameters with strong effects on the response of the model) have been performed against individual and all available experimental data using a Bayesian approach based on the Markov chain Monte Carlo-Metropolis Hastings algorithm. Then, the resulting parameters' uncertainties have been propagated to the uncertainty in model results through a forward model analysis. An empirical relationship for matrix/precipitate interfacial energy in terms of aging temperature and nominal composition has been introduced according to the values of interfacial energy obtained from the model calibration with each given experimental data individually. Using this equation for interfacial energy in the model, all experimental data has been used together to calibrate the other influential parameters in the model. After the probabilistic calibration of the model, the discrepancies between the model outcomes and the experimental data were the main reasons to apply co-Kriging surrogate modeling that takes advantage of an error correlation-based fusion of the model results and the experimental data for more precise prediction of precipitation behavior in this system. [ABSTRACT FROM AUTHOR]