Markov Chain Monte Carlo Algorithm Research Articles

Bounding the speedup of the quantum-enhanced Markov-chain Monte Carlo algorithm

Bounding the speedup of the quantum-enhanced Markov-chain Monte Carlo algorithm

Regularized Bayesian algorithms for Q-matrix inference based on saturated cognitive diagnosis modelling.

Q-matrices are crucial components of cognitive diagnosis models (CDMs), which are used to provide diagnostic information and classify examinees according to their attribute profiles. The absence of an appropriate Q-matrix that correctly reflects item-attribute relationships often limits the widespread use of CDMs. Rather than relying on expert judgment for specification and post-hoc methods for validation, there has been a notable shift towards Q-matrix estimation by adopting Bayesian methods. Nevertheless, their dependency on Markov chain Monte Carlo (MCMC) estimation requires substantial computational burdens and their exploratory tendency is unscalable to large-scale settings. As a scalable and efficient alternative, this study introduces the partially confirmatory framework within a saturated CDM, where the Q-matrix can be partially defined by experts and partially inferred from data. To address the dual needs of accuracy and efficiency, the proposed framework accommodates two estimation algorithms-an MCMC algorithm and a Variational Bayesian Expectation Maximization (VBEM) algorithm. This dual-channel approach extends the model's applicability across a variety of settings. Based on simulated and real data, the proposed framework demonstrated its robustness in Q-matrix inference.

Lifted TASEP: A Solvable Paradigm for Speeding up Many-Particle Markov Chains

Virtually all Markov-chain Monte Carlo algorithms used for sampling a given distribution are reversible, and they satisfy the detailed-balance condition. For local chains, this leads to a slow, diffusive exploration of sample space. Significant speedups can be achieved through nonreversible algorithms with the given distribution as a targeted steady state. However, nonreversible algorithms for sampling are difficult to set up and to analyze, and exact speedup results for interacting many-particle systems are very rare. Here, we introduce the “lifted” totally asymmetric simple exclusion process (TASEP) as an exactly solvable paradigm for nonreversible many-particle Markov chains. It samples the same hard-sphere distribution as the Metropolis algorithm for symmetrically diffusing hard-core particles on a one-dimensional lattice. We solve the lifted TASEP by an unusual kind of coordinate Bethe ansatz and show that it exhibits polynomial (in particle number) speedups in the relaxation time for the asymptotic approach of the steady state, as well as the nonasymptotic mixing time, compared to both Metropolis and Kardar-Parisi-Zhang-based dynamics. The lifted TASEP is the reduction onto the one-dimensional lattice of the successful hard-sphere event-chain Monte Carlo algorithm, and we discuss that it can likewise be generalized to soft interaction potentials. Published by the American Physical Society 2024

A hybrid inversion algorithm to obtain the resistivity of the uninvaded zone based on the array induction log

This study investigates the impact of the drilling mud invasion on the borehole-measured resistivity. The primary objective is to retrieve the true resistivity of the formation, which helps in identifying different fluids in the reservoir. To achieve this goal, We proposed a hybrid inversion approach integrating the Levenberg-Marquardt and Markov Chain Monte Carlo algorithms with a five-parameter formation resistivity model. Synthetic and real-world data are utilized to assess the method's robustness and reliability. The simulated result indicated that the method is reliable when the data noise level is less than 5%.The method applied to real-world data revealed that the resistivity profile on the water zone showed a slight increase in the inverted resistivity from measured resistivity. Meanwhile, in the oil zone, the calculated resistivity revealed a high deviation from the measured resistivity, indicating the effects of mud invasion. The introduced methods are only applicable when the invasions of mud occur within the range of the logging tool's depth of investigation. Moreover, the method may give no reliable result when the invasion exceeds the tool's investigation depth. It indicates its limitation.

Generation of root zone soil moisture from the integration of an all-weather satellite surface soil moisture estimates and an analytical model: A preliminary result in China

In addition to surface soil moisture (SSM) data, root zone soil moisture (RZSM) is particularly significant for irrigation decision-making and agricultural drought warnings since water uptake generally occurs through the root systems of plants. In this study, a daily/0.05° RZSM dataset covering the major land areas of China was generated from the synergistic use of satellite-derived SSM estimates and a RZSM-SSM analytical model. First, based on the SSM and RZSM datasets generated from the fifth generation of atmospheric reanalysis (ERA5-Land), the Markov Chain Monte Carlo algorithm was applied to optimize the soil moisture analytical relationship (SMAR) model to determine four essential parameters pixel-to-pixel over the study area. In addition, the optimized SMAR parameters were integrated with satellite-derived daily/0.05° SSM to predict the daily/0.05° RZSM in China. Finally, due to the lack of in situ RZSM measurements ground truth at a spatial resolution of 0.05°, two RZSM products, namely the Soil Moisture Active Passive (SMAP)-based RZSM and the China Meteorological Administration Land Data Assimilation System (CLDAS)-based RZSM, were used as references to preliminarily assess the estimated data. The results showed overall fair correlations between SMAP_RZSM and estimated RZSM (correlation coefficient R varying from 0.557 to 0.655), and between CLDAS_RZSM and estimated RZSM (correlation coefficient R varying from 0.496 to 0.596) over four typical days representing spring, summer, autumn, and winter in 2019. Although the proposed approach may overestimate or underestimate RZSM compared to the two referenced products in different seasons, an overall acceptable accuracy with unbiased root mean square error of ∼ 0.070–0.090 cm3/cm3 was obtained.

Heterogeneous Mediation Analysis for Cox Proportional Hazards Model With Multiple Mediators.

This study proposes a heterogeneous mediation analysis for survival data that accommodates multiple mediators and sparsity of the predictors. We introduce a joint modeling approach that links the mediation regression and proportional hazards models through Bayesian additive regression trees with shared typologies. The shared tree component is motivated by the fact that confounders and effect modifiers on the causal pathways linked by different mediators often overlap. A sparsity-inducing prior is incorporated to capture the most relevant confounders and effect modifiers on different causal pathways. The individual-specific interventional direct and indirect effects are derived on the scale of the logarithm of hazards and survival function. A Bayesian approach with an efficient Markov chain Monte Carlo algorithm is developed to estimate the conditional interventional effects through the Monte Carlo implementation of the mediation formula. Simulation studies are conducted to verify the empirical performance of the proposed method. An application to the ACTG175 study further demonstrates the method's utility in causal discovery and heterogeneity quantification.

Sampling random graphs with specified degree sequences

The configuration model is a standard tool for uniformly generating random graphs with a specified degree sequence, and is often used as a null model to evaluate how much of an observed network’s structure can be explained by its degree structure alone. A Markov chain Monte Carlo (MCMC) algorithm, based on a degree-preserving double-edge swap, provides an asymptotic solution to sample from the configuration model. However, accurately and efficiently detecting when this Markov chain is sufficiently close to its stationary distribution remains an unsolved problem. Here, we provide a solution to sample from the configuration model using this standard MCMC algorithm. We develop an algorithm, based on the assortativity of the sampled graphs, for estimating the gap between effectively independent MCMC states, and a computationally efficient gap-estimation heuristic derived from analyzing a corpus of 509 empirical networks. We provide a convergence detection method based on the Dickey-Fuller Generalized Least Squares test, which we show is more accurate and efficient than three alternative Markov chain convergence tests. Supplementary materials for the proposed methods can be found here.

Statistical analysis for progressive stress accelerated life test with transmutation Weibull distribution

The Transmutation Weibull distribution is derived from the quadratic rank transmutation map of the Weibull distribution. This distribution integrates the high adaptability of the Weibull distribution in engineering and materials fields, and has great potential utility for modeling lifetime data. Therefore, the article explores the progressive stress accelerated life test model for components under the Transmutation Weibull distribution, conducts statistical analysis using the maximum likelihood estimation (MLE), Lindley’s approximation, and Markov Chain Monte Carlo (MCMC) algorithm to provide approximate estimates for the model. Finally, by comparing the results of numerical simulations, it is confirmed that the Bayesian estimation is more accurate and effective.

Bayesian estimation of stochastic volatility jump diffusion model parameters using S&P 500 and VIX data

The Stochastic Volatility Jump Diffusion (SVJD) model is a common tool used for option pricing. Existing literature has employed many techniques in order to estimate the parameters of this model, including both Bayesian and frequentist approaches. In 2010, Duan and Yeh [Jump and volatility risk premiums implied by VIX. J Econ Dyn Control. 2010;34(11):2232–2244] used a frequentist approach with maximum likelihood estimation that included the VIX, the Chicago Board Options Exchange (CBOE) Volatility Index, which carries real time S&P 500 option price information in addition to the S&P 500 data. In this paper, we combine the previous Bayesian approach in Cape et al. [Estimating Heston's and Bates' models parameters using Markov Chain Monte Carlo simulation. J Stat Comput Simul. 2015;85(11):2295–2314] with the inclusion of VIX by Duan and Yeh. We provide the derivations of the conditional posterior distributions of the SVJD model's parameters and a guide to a Markov Chain Monte Carlo (MCMC) algorithm used to estimate the parameters. We apply the algorithm to the historical S&P 500 and VIX data and provide the subsequent parameter estimates. We offer these tools for others performing similar research with our R code available upon request.

2D Transdimensional Joint Inversion of Radio Magnetotelluric and Electrical Resistivity Tomography Data

Summary The joint inversion of radio magnetotelluric and electrical resistivity tomography data has the potential to reduce the uncertainties in the subsurface conductivity model. This is particularly beneficial when the datasets offer complementary information about the subsurface. However, the traditional gradient-based inversion methods pose challenges in quantifying uncertainty, as they yield a single model with limited appraisal of parameter uncertainty. The Bayesian inversion approach stands out for its capacity to provide quantitative assessments of uncertainty in the inverted model parameters. This is accomplished by generating an ensemble of models, leading to a posterior distribution that encapsulates both prior information concerning model parameters and the dataset information. We have implemented a transdimensional Markov chain Monte Carlo algorithm to perform the joint inversion of radio magnetotelluric and electrical resistivity tomography data. Through synthetic data studies, we illustrate how the inclusion of two complementary datasets can effectively reduce uncertainties in model parameters and how the model parameter uncertainties can be quantified. Subsequently, the developed algorithm is tested using exemplary field data from a waste site near Roorkee, India. Intensive prior geoelectric and transient electromagnetic as well as radio magnetotelluric studies investigated possible waste water seepage with a potential to contaminate the shallow aquifers. The derived subsurface structure from our transdimensional Bayesian results compare well with the deterministic results for the exemplary profile, but in addition provide comprehensive uncertainty estimates.

MCMC stopping rules in latent variable modelling.

Bayesian analysis relies heavily on the Markov chain Monte Carlo (MCMC) algorithm to obtain random samples from posterior distributions. In this study, we compare the performance of MCMC stopping rules and provide a guideline for determining the termination point of the MCMC algorithm in latent variable models. In simulation studies, we examine the performance of four different MCMC stopping rules: potential scale reduction factor (PSRF), fixed-width stopping rule, Geweke's diagnostic, and effective sample size. Specifically, we evaluate these stopping rules in the context of the DINA model and the bifactor item response theory model, two commonly used latent variable models in educational and psychological measurement. Our simulation study findings suggest that single-chain approaches outperform multiple-chain approaches in terms of item parameter accuracy. However, when it comes to person parameter estimates, the effect of stopping rules diminishes. We caution against relying solely on the univariate PSRF, which is the most popular method, as it may terminate the algorithm prematurely and produce biased item parameter estimates if the cut-off value is not chosen carefully. Our research offers guidance to practitioners on choosing suitable stopping rules to improve the precision of the MCMC algorithm in models involving latent variables.

A landslide runout model for sediment transport, landscape evolution, and hazard assessment applications

Abstract. We developed a new rule-based, cellular-automaton algorithm for predicting the hazard extent, sediment transport, and topographic change associated with the runout of a landslide. This algorithm, which we call MassWastingRunout (MWR), is coded in Python and implemented as a component for the package Landlab. MWR combines the functionality of simple runout algorithms used in landscape evolution and watershed sediment yield models with the predictive detail typical of runout models used for landslide inundation hazard mapping. An initial digital elevation model (DEM), a regolith depth map, and the location polygon of the landslide source area are the only inputs required to run MWR to model the entire runout process. Runout relies on the principle of mass conservation and a set of topographic rules and empirical formulas that govern erosion and deposition. For the purpose of facilitating rapid calibration to a site, MWR includes a calibration utility that uses an adaptive Bayesian Markov chain Monte Carlo algorithm to automatically calibrate the model to match observed runout extent, deposition, and erosion. Additionally, the calibration utility produces empirical probability density functions of each calibration parameter that can be used to inform probabilistic implementation of MWR. Here we use a series of synthetic terrains to demonstrate basic model response to topographic convergence and slope, test calibrated model performance relative to several observed landslides, and briefly demonstrate how MWR can be used to develop a probabilistic runout hazard map. A calibrated runout model may allow for region-specific and more insightful predictions of landslide impact on landscape morphology and watershed-scale sediment dynamics and should be further investigated in future modeling studies.

MCMC for Bayesian nonparametric mixture modeling under differential privacy

Estimating the probability density of a population while preserving the privacy of individuals in that population is an important and challenging problem that has received considerable attention in recent years. While the previous literature focused on frequentist approaches, in this paper, we propose a Bayesian nonparametric mixture model under differential privacy (DP) and present two Markov chain Monte Carlo (MCMC) algorithms for posterior inference. One is a marginal approach, resembling Neal’s algorithm 5 with a pseudo-marginal Metropolis-Hastings move, and the other is a conditional approach. Although our focus is primarily on local DP, we show that our MCMC algorithms can be easily extended to deal with global differential privacy mechanisms. Moreover, for some carefully chosen mechanisms and mixture kernels, we show how auxiliary parameters can be analytically marginalized, allowing standard MCMC algorithms (i.e., non-privatized, such as Neal’s Algorithm 2) to be efficiently employed. Our approach is general and applicable to any mixture model and privacy mechanism. In several simulations and a real case study, we discuss the performance of our algorithms and evaluate different privacy mechanisms proposed in the frequentist literature.

Gradient boosting: A computationally efficient alternative to Markov chain Monte Carlo sampling for fitting large Bayesian spatio-temporal binomial regression models

Disease forecasting and surveillance often involve fitting models to a tremendous volume of historical testing data collected over space and time. Bayesian spatio-temporal regression models fit with Markov chain Monte Carlo (MCMC) methods are commonly used for such data. When the spatio-temporal support of the model is large, implementing an MCMC algorithm becomes a significant computational burden. This research proposes a computationally efficient gradient boosting algorithm for fitting a Bayesian spatio-temporal mixed effects binomial regression model. We demonstrate our method on a disease forecasting model and compare it to a computationally optimized MCMC approach. Both methods are used to produce monthly forecasts for Lyme disease, anaplasmosis, ehrlichiosis, and heartworm disease in domestic dogs for the contiguous United States. The data have a spatial support of 3108 counties and a temporal support of 108–138 months with 71–135 million test results. The proposed estimation approach is several orders of magnitude faster than the optimized MCMC algorithm, with a similar mean absolute prediction error.

A Bayesian hybrid method for the analysis of generalized linear models with missing-not-at-random covariates

Missing data handling is one of the main problems in modelling, particularly if the missingness is of type missing-not-at-random (MNAR) where missingness occurs due to the actual value of the observation. The focus of the current article is generalized linear modelling of fully observed binary response variables depending on at least one MNAR covariate. For the traditional analysis of such models, an individual model for the probability of missingness is assumed and incorporated in the model framework. However, this probability model is untestable, as the missingness of MNAR data depend on their actual values that would have been observed otherwise. In this article, we consider creating a model space that consist of all possible and plausible models for probability of missingness and develop a hybrid method in which a reversible jump Markov chain Monte Carlo (RJMCMC) algorithm is combined with Bayesian Model Averaging (BMA). RJMCMC is adopted to obtain posterior estimates of model parameters as well as probability of each model in the model space. BMA is used to synthesize coefficient estimates from all models in the model space while accounting for model uncertainties. Through a validation study with a simulated data set and a real data application, the performance of the proposed methodology is found to be satisfactory in accuracy and efficiency of estimates.

Efficient Modeling of Spatial Extremes over Large Geographical Domains

Various natural phenomena exhibit spatial extremal dependence at short spatial distances. However, existing models proposed in the spatial extremes literature often assume that extremal dependence persists across the entire domain. This is a strong limitation when modeling extremes over large geographical domains, and yet it has been mostly overlooked in the literature. We here develop a more realistic Bayesian framework based on a novel Gaussian scale mixture model, with the Gaussian process component defined by a stochastic partial differential equation yielding a sparse precision matrix, and the random scale component modeled as a low-rank Pareto-tailed or Weibull-tailed spatial process determined by compactly-supported basis functions. We show that our proposed model is approximately tail-stationary and that it can capture a wide range of extremal dependence structures. Its inherently sparse structure allows fast Bayesian computations in high spatial dimensions based on a customized Markov chain Monte Carlo algorithm prioritizing calibration in the tail. We fit our model to analyze heavy monsoon rainfall data in Bangladesh. Our study shows that our model outperforms natural competitors and that it fits precipitation extremes well. We finally use the fitted model to draw inference on long-term return levels for marginal precipitation and spatial aggregates.

Analysis of dependent complementary competing risks data from a generalized inverted family of lifetime distributions under a maximum ranked set sampling procedure with unequal samples

This paper explores analysis of a dependent complementary competing risks model when the failure causes are distributed by the proposed generalized inverted family of lifetime distributions. Under maximum ranked set sampling with unequal samples (MRSSU), statistical inference of model parameters and reliability indices is discussed under classical frequentist and Bayesian approaches, respectively. Maximum likelihood estimators along with their existence and uniqueness are obtained for model parameters, and associated approximate confidence intervals are constructed in consequence. Bayesian estimation is also performed with respect to general flexible priors, and the Markov Chain Monte Carlo (MCMC) algorithm is proposed for complex posterior computation. The study further examines classical and Bayesian estimations with order restriction of parameters when additional historical information is available in the MRSSU scenario. Finally, the performance of different results is evaluated through numerical simulations and a real data example is presented for demonstrating the application of our methods.

School desegregation by redrawing district boundaries.

Schools in the United States remain heavily segregated by race and income. Previous work demonstrates districts can promote group diversity within their schools with policies like redrawing attendance zones. Yet, the promise of such policies in many areas is limited by the fact that most school segregation occurs between school districts, and not between schools in the same district. I adapt Markov Chain Monte Carlo algorithms from legislative redistricting to redraw school district boundaries that decrease segregation while maintaining desirable criteria like distance to school and using only existing school facilities. Focusing on New Jersey, where the segregation of Black and Hispanic students from White and Asian students is among the worst in the country, I demonstrate that redrawing school districts could reduce more than 40% of existing segregation in the median New Jersey county, compared to less than 5% for redrawing attendance zones alone. Finally, I show how my proposed methodology can be applied to as few as two districts to reduce segregation in proposed consolidations, when small districts are merged into a larger district.

Multivariate VAR system of regional economic growth under big data

Traditional research has problems such as limited data samples, limited variable selection, and insufficient real-time analysis capabilities. With the advent of the big data era, researchers are increasingly using big data analysis to discover the laws of development. This paper combines the VAR system with big data analysis technology to study regional economic growth variables, simulate economic changes in region A, and use the Markov chain Monte Carlo algorithm and historical simulation method to analyze the process of industrial change. The model accuracy of the two algorithms is evaluated by the historical data observation loss value of the VAR post-test. The results show that at a confidence level of 0.95, the risk prediction of the historical simulation method is more accurate. The study also found that adding additional variables to the VAR system will lead to model information loss and an increase in parameters.

Normalizing flow sampling with Langevin dynamics in the latent space

AbstractNormalizing flows (NF) use a continuous generator to map a simple latent (e.g. Gaussian) distribution, towards an empirical target distribution associated with a training data set. Once trained by minimizing a variational objective, the learnt map provides an approximate generative model of the target distribution. Since standard NF implement differentiable maps, they may suffer from pathological behaviors when targeting complex distributions. For instance, such problems may appear for distributions on multi-component topologies or characterized by multiple modes with high probability regions separated by very unlikely areas. A typical symptom is the explosion of the Jacobian norm of the transformation in very low probability areas. This paper proposes to overcome this issue thanks to a new Markov chain Monte Carlo algorithm to sample from the target distribution in the latent domain before transporting it back to the target domain. The approach relies on a Metropolis adjusted Langevin algorithm whose dynamics explicitly exploits the Jacobian of the transformation. Contrary to alternative approaches, the proposed strategy preserves the tractability of the likelihood and it does not require a specific training. Notably, it can be straightforwardly used with any pre-trained NF network, regardless of the architecture. Experiments conducted on synthetic and high-dimensional real data sets illustrate the efficiency of the method.