Markov Chain Monte Carlo Iterations Research Articles

Bayesian Structural Time Series Model for Forecasting the Composite Stock Price Index in Indonesia

One of the models that can be used to predict time series data is the Bayesian Structural Time Series (BSTS) model. The BSTS model is a more modern model and can handle data movement better. In the BSTS model, the Markov Chain Monte Carlo (MCMC) sampling algorithm is used to simulate the posterior distribution, which smoothes the forecasting results over a large number of potential models using Bayesian averaging models. The purpose of this study was to obtain the best BSTS model for Composite Stock Price Index (CSPI) data in Indonesia based on the state component and the number of MCMC iterations, and obtain forecasting results for CSPI value in Indonesia for the next 24 months, namely the period July 2023 to June 2024. The results obtained are based on a comparison of the R-square values in the model, the BSTS model with local linear trend and seasonal state components, and the number of MCMC iterations n = 5 00 is the best BSTS model that can be used for forecasting the CSPI value in Indonesia with an R-square value of 99.96%. The results of forecasting the CSPI value in Indonesia for the period July 2023 to June 2024 range from 6589 to 6760, with the lowest forecasting value in October 2023 and the highest in March 2023.

Assessing geometrical uncertainties in geological interface models using Markov chain Monte Carlo sampling via abstract graph

Uncertainty assessment is a common requirement in 3D modeling applications. The Markov chain Monte-Carlo (MCMC) method is a practical way to generate multiple realizations of 3D models to evaluate model uncertainty. However, when probing high-dimensional target distributions related to the detailed geometry of 3D geological interfaces, the generating 3D models by existing MCMC methods always suffer from the presence of artifacts in large geometrical deformation that modeling deformed geometries in stochastic perturbation, which limits their effectiveness and efficiency. This paper presents an abstract graph-based MCMC method for the uncertainty assessment of 3D geological interfaces. An abstract graph is used to represent the reasonable shape of the geological interface. In proposing new candidate 3D models via the MCMC iterations, the 3D models are perturbed such that the energy function measuring the distortion of the abstract graph is minimized in a least-squares sense. This can preserve the plausible geometry of the 3D models against high perturbations and provide artifact-free 3D models as candidates for MCMC sampling. The resulting MCMC iterations enable the drawing of probable realizations of 3D models while maintaining a high acceptance rate in probing the complicated target distributions of 3D geological interface models. The proposed method was applied to the assessment of uncertainties in three regional faults in the Jiaodong Peninsula, eastern China. The uncertainty assessment was carried out based on a complex probability distribution with consideration of geological-geophysical observations and prior knowledge of fault geometry. The results show that the proposed MCMC method could reasonably locate the uncertainties in free-form fault models defined by thousands of constraints. And the abstract graph-based MCMC method can sample highly-perturbed and geometrically plausible 3D models in high-resolution, which is crucial to the effectiveness and efficiency of the geometrical uncertainty assessment.

A semiparametric Bayesian method with birth-death Markov Chain Monte Carlo algorithm for extreme mooring tension analysis

A series of 1: 50 model tests have been conducted to study the dynamics of a taut mooring system and two hybrid mooring systems attached to a deep-water semi-submersible in the 100-year wave condition of the South China Sea. Three typical environmental conditions were tested, including head sea, beam sea and quartering sea. The mixture Gamma- Generalized Pareto distribution (GPD) model is applied to study short term extreme dynamic mooring tensions based on the measured data, the results are compared with the estimations from mixture Gamma and two-parameter Weibull models. A birth-death Markov chain Monte Carlo (MCMC) sampling procedure is developed to fit the parameters of a mixture Gamma- GPD via Bayesian inference, and an extensive discussion of the de-clustering procedure, parameters of the prior distribution for threshold and MCMC iteration steps on Bayesian inference are carried out. It is found that the Weibull model underestimates the most probable largest dynamic mooring tensions. On the contrary, the mixture Gamma model overestimates the extreme dynamic mooring tension significantly. The mixture Gamma- GPD can present good estimations of extreme dynamic mooring tension, and its performance is robust to the sample definition, MCMC iteration steps and parameters of threshold prior distribution.

On the identifiability of Bayesian factor analytic models

A well known identifiability issue in factor analytic models is the invariance with respect to orthogonal transformations. This problem burdens the inference under a Bayesian setup, where Markov chain Monte Carlo (MCMC) methods are used to generate samples from the posterior distribution. We introduce a post-processing scheme in order to deal with rotation, sign and permutation invariance of the MCMC sample. The exact version of the contributed algorithm requires to solve 2^q assignment problems per (retained) MCMC iteration, where q denotes the number of factors of the fitted model. For large numbers of factors two approximate schemes based on simulated annealing are also discussed. We demonstrate that the proposed method leads to interpretable posterior distributions using synthetic and publicly available data from typical factor analytic models as well as mixtures of factor analyzers. An R package is available online at CRAN web-page.

Double Happiness: Enhancing the Coupled Gains of L-lag Coupling via Control Variates

The recently proposed L-lag coupling for unbiased Markov chain Monte Carlo (MCMC) calls for a joint celebration by MCMC practitioners and theoreticians. For practitioners, it circumvents the thorny issue of deciding the burn-in period or when to terminate an MCMC sampling process, and opens the door for safe parallel implementation. For theoreticians, it provides a powerful tool to establish elegant and easily estimable bounds on the exact error of an MCMC approximation at any finite number of iterates. A serendipitous observation about the bias-correcting term leads us to introduce naturally available control variates into the L-lag coupling estimators. In turn, this extension enhances the coupled gains of L-lag coupling, because it results in more efficient unbiased estimators, as well as a better bound on the total variation error of MCMC iterations, albeit the gains diminish as L increases. Specifically, the new upper bound is theoretically guaranteed to never exceed the one given previously. We also argue that L-lag coupling represents a coupling for the future, breaking from the coupling-from-the-past type of perfect sampling, by reducing the generally unachievable requirement of being perfect to one of being unbiased, a worthwhile trade-off for ease of implementation in most practical situations. The theoretical analysis is supported by numerical experiments that show tighter bounds and a gain in efficiency when control variates are introduced.

Parameter Estimation of Partially Observed Turbulent Systems Using Conditional Gaussian Path-Wise Sampler

Parameter estimation of complex nonlinear turbulent dynamical systems using only partially observed time series is a challenging topic. The nonlinearity and partial observations often impede using closed analytic formulae to recover the model parameters. In this paper, an exact path-wise sampling method is developed, which is incorporated into a Bayesian Markov chain Monte Carlo (MCMC) algorithm in light of data augmentation to efficiently estimate the parameters in a rich class of nonlinear and non-Gaussian turbulent systems using partial observations. This path-wise sampling method exploits closed analytic formulae to sample the trajectories of the unobserved variables, which avoid the numerical errors in the general sampling approaches and significantly increase the overall parameter estimation efficiency. The unknown parameters and the missing trajectories are estimated in an alternating fashion in an adaptive MCMC iteration algorithm with rapid convergence. It is shown based on the noisy Lorenz 63 model and a stochastically coupled FitzHugh–Nagumo model that the new algorithm is very skillful in estimating the parameters in highly nonlinear turbulent models. The model with the estimated parameters succeeds in recovering the nonlinear and non-Gaussian features of the truth, including capturing the intermittency and extreme events, in both test examples.

The Block-Poisson Estimator for Optimally Tuned Exact Subsampling MCMC

Speeding up Markov chain Monte Carlo (MCMC) for datasets with many observations by data subsampling has recently received considerable attention. A pseudo-marginal MCMC method is proposed that estimates the likelihood by data subsampling using a block-Poisson estimator. The estimator is a product of Poisson estimators, allowing us to update a single block of subsample indicators in each MCMC iteration so that a desired correlation is achieved between the logs of successive likelihood estimates. This is important since pseudo-marginal MCMC with positively correlated likelihood estimates can use substantially smaller subsamples without adversely affecting the sampling efficiency. The block-Poisson estimator is unbiased but not necessarily positive, so the algorithm runs the MCMC on the absolute value of the likelihood estimator and uses an importance sampling correction to obtain consistent estimates of the posterior mean of any function of the parameters. Our article derives guidelines to select the optimal tuning parameters for our method and shows that it compares very favorably to regular MCMC without subsampling, and to two other recently proposed exact subsampling approaches in the literature. Supplementary materials for this article are available online.

Bayesian analysis of short term extreme mooring tension for a point absorber with mixture of gamma and generalised pareto distributions

Model tests have been conducted to study the performance of slack, catenary and hybrid mooring systems for a point absorber in survival sea states. The snap loads can be found in mooring tension time series of all the tested mooring configurations. The classical Weibull distribution is applied to model the extreme dynamic mooring tensions. In addition to the Weibull distribution, the Bayesian inference method with a new Markov chain Monte Carlo sampling procedure is proposed to study the short term extreme mooring tensions based on model test results. Before applying this method to analyse extreme mooring tension, its accuracy is validated. The probability densities as well as short term extreme dynamic tension estimated by a mixture of Gamma and Generalised Pareto Distributions models are compared with the two-parameter Weibull model results. In the end, an extensive discussion of about the mixture of Gamma components, peak definitions, threshold and Markov chain Monte Carlo iteration on Bayesian inference is carried out. It is validated that the mixture model proposed in this paper shows good performance in predicting the extreme mooring tensions even when the snap loads occur, while the Weibull distribution fails to give accuracy predictions.

Most Likely Optimal Subsampled Markov Chain Monte Carlo

Markov Chain Monte Carlo (MCMC) requires to evaluate the full data likelihood at different parameter values iteratively and is often computationally infeasible for large data sets. This paper proposes to approximate the log-likelihood with subsamples taken according to nonuniform subsampling probabilities, and derives the most likely optimal (MLO) subsampling probabilities for better approximation. Compared with existing subsampled MCMC algorithm with equal subsampling probabilities, the MLO subsampled MCMC has a higher estimation efficiency with the same subsampling ratio. The authors also derive a formula using the asymptotic distribution of the subsampled log-likelihood to determine the required subsample size in each MCMC iteration for a given level of precision. This formula is used to develop an adaptive version of the MLO subsampled MCMC algorithm. Numerical experiments demonstrate that the proposed method outperforms the uniform subsampled MCMC.

Wang-Landau algorithm as stochastic optimization and its acceleration.

We show that the Wang-Landau algorithm can be formulated as a stochastic gradient descent algorithm minimizing a smooth and convex objective function, of which the gradient is estimated using Markov chain Monte Carlo iterations. The optimization formulation provides us another way to establish the convergence rate of the Wang-Landau algorithm, by exploiting the fact that almost surely the density estimates (on the logarithmic scale) remain in a compact set, upon which the objective function is strongly convex. The optimization viewpoint motivates us to improve the efficiency of the Wang-Landau algorithm using popular tools including the momentum method and the adaptive learning rate method. We demonstrate the accelerated Wang-Landau algorithm on a two-dimensional Ising model and a two-dimensional ten-state Potts model.

Estimating red noise in quasi-periodic signals with MCMC-based Bayesian

Multi-parameter Bayesian inferences based on Markov chain Monte Carlo (MCMC) samples have been widely used to estimate red noise in solar period-periodic signals. To MCMC, proper priors and sufficient iterations are prerequisites ensuring the accuracy of red noise estimation. We used MCMC-based Bayesian inferences to estimate 100 groups of red noise synthesized randomly for evaluating its accuracy. At the same time, the Brooks-Gelman algorithm was employed to precisely diagnose the convergence of the Markov chains generated by MCMC. The root-mean-square error of parameter inferences to the synthetic data is only 1.14. Furthermore, we applied the algorithm to analyze the oscillation modes in a sunspot and a flare. A 70 s period is detected in the sunspot umbra in addition to 3- and 5-minute periods, and a 40 s period is detected in the flare. The results prove that estimating red noise with MCMC-based Bayesian has more high accuracy in the case of proper priors and convergence. We also find that the number of iterations increases dramatically to achieve convergence as the number of parameters grows. Therefore, we strongly recommend that when estimating red noise with MCMC-based Bayesian, different initial values must be selected to ensure that the entire posterior distribution is covered.

What Is Appropriate Upper Endoscopic Interval Among Dyspeptic Patients With Previously Normal Endoscopy? A Multicenter Study With Bayesian Change Point Analysis.

Background/AimsAppropriate interval for performing follow-up endoscopy among dyspeptic patients without abnormal findings on previous endoscopy is unclear. We analyzed the multicenter-collected data from the Korean Society of Neurogastroenterology and Motility.MethodsWe collected clinical data of the patients who visited the gastroenterology department and underwent 2 or more sessions of upper endoscopy during 2012–2017 at 6 university hospitals in Korea. Patients with endoscopic interval between 90 days and 760 days were included. For those with multiple endoscopic sessions, only the first 2 were analyzed. Positive outcome was defined as adenoma or cancer in the upper gastrointestinal tract. To identify the point of change and estimate the properties of the stochastic process before and after the change, we used Bayesian regression with Metropolis-Hastings algorithm.ResultsThere were 1595 patients. Mean age was 58.8 years (standard deviation, 12.8). Median interval of endoscopy was 437 days (standard deviation, 153). On follow-up endoscopy, there were 12 patients (0.75%) who had neoplasia (4 with gastric cancer and 8 with gastric adnoma). As with the prior hypothesis, we presumed the change point (CP) of increase in frequency of organic lesion as 360 days. After random-walk Metropolis-Hastings sampling with Markov-Chain Monte Carlo iterations of 5000, the CP was 560 days (95% credible interval, 139–724). Estimated average of frequency of dysplastic lesions increased by a factor of 4.4 after the estimated CP.ConclusionTo rule out dysplastic lesions among dyspeptic patients who had previously normal endoscopy, a 2-year interval could be offered as follow-up interval for repeat upper endoscopy.

Dynamically Updated Spatially Varying Parameterizations of Hierarchical Bayesian Models for Spatial Data

ABSTRACTFitting hierarchical Bayesian models to spatially correlated datasets using Markov chain Monte Carlo (MCMC) techniques is computationally expensive. Complicated covariance structures of the underlying spatial processes, together with high-dimensional parameter space, mean that the number of calculations required grows cubically with the number of spatial locations at each MCMC iteration. This necessitates the need for efficient model parameterizations that hasten the convergence and improve the mixing of the associated algorithms. We consider partially centred parameterizations (PCPs) which lie on a continuum between what are known as the centered (CP) and noncentered parameterizations (NCP). By introducing a weight matrix we remove the conditional posterior correlation between the fixed and the random effects, and hence construct a PCP which achieves immediate convergence for a three-stage model, based on multiple Gaussian processes with known covariance parameters. When the covariance parameters are unknown we dynamically update the parameterization within the sampler. The PCP outperforms both the CP and the NCP and leads to a fully automated algorithm which has been demonstrated in two simulation examples. The effectiveness of the spatially varying PCP is illustrated with a practical dataset of nitrogen dioxide concentration levels. Supplemental materials consisting of appendices, datasets, and computer code to reproduce the results are available online.

A frequentist approach to Bayesian asymptotics

Ergodic theorem shows that ergodic averages of the posterior draws converge in probability to the posterior mean under the stationarity assumption. The literature also shows that the posterior distribution is asymptotically normal when the sample size of the original data considered goes to infinity. To the best of our knowledge, there is little discussion on the large sample behaviour of the posterior mean. In this paper, we aim to fill this gap. In particular, we extend the posterior mean idea to the conditional mean case, which is conditioning on a given vector of summary statistics of the original data. We establish a new asymptotic theory for the conditional mean estimator for the case when both the sample size of the original data concerned and the number of Markov chain Monte Carlo iterations go to infinity. Simulation studies show that this conditional mean estimator has very good finite sample performance. In addition, we employ the conditional mean estimator to estimate a GARCH(1,1) model for S&P 500 stock returns and find that the conditional mean estimator performs better than quasi-maximum likelihood estimation in terms of out-of-sample forecasting.

Revisiting random walk based sampling in networks: evasion of burn-in period and frequent regenerations

BackgroundIn the framework of network sampling, random walk (RW) based estimation techniques provide many pragmatic solutions while uncovering the unknown network as little as possible. Despite several theoretical advances in this area, RW based sampling techniques usually make a strong assumption that the samples are in stationary regime, and hence are impelled to leave out the samples collected during the burn-in period.MethodsThis work proposes two sampling schemes without burn-in time constraint to estimate the average of an arbitrary function defined on the network nodes, for example, the average age of users in a social network. The central idea of the algorithms lies in exploiting regeneration of RWs at revisits to an aggregated super-node or to a set of nodes, and in strategies to enhance the frequency of such regenerations either by contracting the graph or by making the hitting set larger. Our first algorithm, which is based on reinforcement learning (RL), uses stochastic approximation to derive an estimator. This method can be seen as intermediate between purely stochastic Markov chain Monte Carlo iterations and deterministic relative value iterations. The second algorithm, which we call the Ratio with Tours (RT)-estimator, is a modified form of respondent-driven sampling (RDS) that accommodates the idea of regeneration.ResultsWe study the methods via simulations on real networks. We observe that the trajectories of RL-estimator are much more stable than those of standard random walk based estimation procedures, and its error performance is comparable to that of respondent-driven sampling (RDS) which has a smaller asymptotic variance than many other estimators. Simulation studies also show that the mean squared error of RT-estimator decays much faster than that of RDS with time.ConclusionThe newly developed RW based estimators (RL- and RT-estimators) allow to avoid burn-in period, provide better control of stability along the sample path, and overall reduce the estimation time. Our estimators can be applied in social and complex networks.

Estimation of sensitivity and specificity of pregnancy diagnosis using transrectal ultrasonography and ELISA for pregnancy-associated glycoprotein in dairy cows using a Bayesian latent class model

AIMS: To determine the sensitivity (Se) and specificity (Sp) of pregnancy diagnosis using transrectal ultrasonography and an ELISA for pregnancy-associated glycoprotein (PAG) in milk, in lactating dairy cows in seasonally calving herds approximately 85–100 days after the start of the herd’s breeding period.METHODS: Paired results were used from pregnancy diagnosis using transrectal ultrasonography and ELISA for PAG in milk carried out approximately 85 and 100 days after the start of the breeding period, respectively, from 879 cows from four herds in Victoria, Australia. A Bayesian latent class model was used to estimate the proportion of cows pregnant, the Se and Sp of each test, and covariances between test results in pregnant and non-pregnant cows. Prior probability estimates were defined using beta distributions for the expected proportion of cows pregnant, Se and Sp for each test, and covariances between tests. Markov Chain Monte Carlo iterations identified posterior distributions for each of the unknown variables. Posterior distributions for each parameter were described using medians and 95% probability (i.e. credible) intervals (PrI). The posterior median estimates for Se and Sp for each test were used to estimate positive predictive and negative predictive values across a range of pregnancy proportions.RESULTS: The estimate for proportion pregnant was 0.524 (95% PrI = 0.485–0.562). For pregnancy diagnosis using transrectal ultrasonography, Se and Sp were 0.939 (95% PrI = 0.890–0.974) and 0.943 (95% PrI = 0.885–0.984), respectively; for ELISA, Se and Sp were 0.963 (95% PrI = 0.919–0.990) and 0.870 (95% PrI = 0.806–0.931), respectively. The estimated covariance between test results was 0.033 (95% PrI = 0.008–0.046) and 0.035 (95% PrI = 0.018–0.078) for pregnant and non-pregnant cows, respectively. Pregnancy diagnosis results using transrectal ultrasonography had a higher positive predictive value but lower negative predictive value than results from the ELISA across the range of pregnancy proportions assessed.CONCLUSIONS AND CLINICAL RELEVANCE: Pregnancy diagnosis using transrectal ultrasonography and ELISA for PAG in milk had similar Se but differed in predictive values. Pregnancy diagnosis in seasonally calving herds around 85–100 days after the start of the breeding period using the ELISA is expected to result in a higher negative predictive value but lower positive predictive value than pregnancy diagnosis using transrectal ultrasonography. Thus, with the ELISA, a higher proportion of the cows with negative results will be non-pregnant, relative to results from transrectal ultrasonography, but a lower proportion of cows with positive results will be pregnant.

On parallelizable Markov chain Monte Carlo algorithms with waste-recycling

Parallelizable Markov chain Monte Carlo (MCMC) generates multiple proposals and parallelizes the evaluations of the likelihood function on different cores at each MCMC iteration. Inspired by Calderhead (Proc Natl Acad Sci 111(49):17408–17413, 2014), we introduce a general ‘waste-recycling’ framework for parallelizable MCMC, under which we show that using weighted samples from waste-recycling is preferable to resampling in terms of both statistical and computational efficiencies. We also provide a simple-to-use criteria, the generalized effective sample size, for evaluating efficiencies of parallelizable MCMC algorithms, which applies to both the waste-recycling and the vanilla versions. A moment estimator of the generalized effective sample size is provided and shown to be reasonably accurate by simulations.

Polynomial Chaos–Based Bayesian Inference of K-Profile Parameterization in a General Circulation Model of the Tropical Pacific

Abstract The authors present a polynomial chaos (PC)–based Bayesian inference method for quantifying the uncertainties of the K-profile parameterization (KPP) within the MIT general circulation model (MITgcm) of the tropical Pacific. The inference of the uncertain parameters is based on a Markov chain Monte Carlo (MCMC) scheme that utilizes a newly formulated test statistic taking into account the different components representing the structures of turbulent mixing on both daily and seasonal time scales in addition to the data quality, and filters for the effects of parameter perturbations over those as a result of changes in the wind. To avoid the prohibitive computational cost of integrating the MITgcm model at each MCMC iteration, a surrogate model for the test statistic using the PC method is built. Because of the noise in the model predictions, a basis-pursuit-denoising (BPDN) compressed sensing approach is employed to determine the PC coefficients of a representative surrogate model. The PC surrogate is then used to evaluate the test statistic in the MCMC step for sampling the posterior of the uncertain parameters. Results of the posteriors indicate good agreement with the default values for two parameters of the KPP model, namely the critical bulk and gradient Richardson numbers; while the posteriors of the remaining parameters were barely informative.

Efficient Technique for Assessing Actual Non‐normal Quality Loss: Markov Chain Monte Carlo

Markov chain Monte Carlo (MCMC) techniques have been extensively developed and are accepted for solving various real‐world problems. However, process capabilities are rarely analyzed with the means of MCMC. This study integrates the MCMC technique into Bayesian models for assessing the well‐known quality loss index Cpm for gamma and Weibull process distributions. After the MCMC iterations are completed, the quality manager can make reliable decisions via the proposed credible intervals. Furthermore, this study provides performance comparisons of the estimators of Cpm obtained by the MCMC and bootstrap techniques. Simulations show that the MCMC technique performs better than the bootstrap technique in most of the cases that were considered. Copyright © 2016 John Wiley & Sons, Ltd.

A Bayesian micro-simulation to evaluate the cost-effectiveness of interventions for mastitis control during the dry period in UK dairy herds

Importance of the dry period with respect to mastitis control is now well established although the precise interventions that reduce the risk of acquiring intramammary infections during this time are not clearly understood. There are very few intervention studies that have measured the clinical efficacy of specific mastitis interventions within a cost-effectiveness framework so there remains a large degree of uncertainty about the impact of a specific intervention and its costeffectiveness. The aim of this study was to use a Bayesian framework to investigate the cost-effectiveness of mastitis controls during the dry period. Data were assimilated from 77 UK dairy farms that participated in a British national mastitis control programme during 2009–2012 in which the majority of intramammary infections were acquired during the dry period. The data consisted of clinical mastitis (CM) and somatic cell count (SCC) records, herd management practices and details of interventions that were implemented by the farmer as part of the control plan. The outcomes used to measure the effectiveness of the interventions were i) changes in the incidence rate of clinical mastitis during the first 30days after calving and ii) the rate at which cows gained new infections during the dry period (measured by SCC changes across the dry period from <200,000cells/ml to >200,000cells/ml). A Bayesian one-step microsimulation model was constructed such that posterior predictions from the model incorporated uncertainty in all parameters. The incremental net benefit was calculated across 10,000 Markov chain Monte Carlo iterations, to estimate the cost-benefit (and associated uncertainty) of each mastitis intervention. Interventions identified as being cost-effective in most circumstances included selecting dry-cow therapy at the cow level, dry-cow rations formulated by a qualified nutritionist, use of individual calving pens, first milking cows within 24h of calving and spreading bedding evenly in dry-cow yards. The results of this study highlighted the efficacy of specific mastitis interventions in UK conditions which, when incorporated into a costeffectiveness framework, can be used to optimize decision making in mastitis control. This intervention study provides an example of how an intuitive and clinically useful Bayesian approach can be used to form the basis of an on-farm decision support tool.