A more efficient Gibbs sampler estimation using steady-state simulation: applications to public health studies

This paper presents an overview of Markov Chain Monte Carlo (MCMC) methods for statistical inference and applications. The article begins by describing ordinary Monte Carlo methods, which in principle has the same goals as the MCMC but can hardly be implemented in practice. Following that basic Markov Chain Monte Carlo is discussed, which is founded on the Hastings algorithm and includes Metropolis method and the Gibbs sampler as special cases. Finally, various special applications of Markov Chain Monte Carlo methods are briefly mentioned and some recent development of MCMC are covered in final remarks section.KeywordsReal EstateMarkov Chain Monte CarloGibbs SamplerDetailed BalanceMarkov Chain Monte Carlo MethodThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

As themany examples in this book illustrate,Markov chainMonte Carlo (MCMC)methods have revolutionized Bayesian statistical analyses. Rather than using off-the-shelf models and methods, we can use MCMC to fit application-specific models that are designed to account for the particular complexities of a problem. These complex multilevel models are becoming more prevalent throughout the natural, social, and engineering sciences largely because of the ease of using standard MCMC methods such as the Gibbs and MetropolisHastings (MH) samplers. Indeed, the ability to easily fit statistical models that directly represent the complexity of a data-generationmechanismhas arguably lead to the increased popularity of Bayesian methods in many scientific disciplines. Although simple standard methods work surprisingly well in many problems, neitherthe Gibbs nor theMH sampler can directly handle problemswith very high posterior correlations among theparameters. Themarginal distribution of a givenparameter ismuchmore variable than the corresponding full conditional distribution in this case, causing the Gibbs sampler to take small steps. With MH a proposal distribution that does not account for the posterior correlation either has far toomuchmass in regions of low posterior probability or has such smallmarginal variances that only small steps are proposed, causing high rejection rates and/or high autocorrelations in the resultingMarkov chains. Unfortunately, accounting for the posterior correlation requires more information about the posterior distribution than is typically available when the proposal distribution is constructed. Much work has been devoted to developing computational methods that extend theusefulness of these standard tools in the presence of high correlations. For Gibbs sampling, for example, it is nowwell known that blocking or grouping steps (Liu et al., 1994), nesting steps (van Dyk, 2000), collapsing or marginalizing parameters (Liu et al., 1994; Meng and vanDyk, 1999), incorporatingauxiliaryvariables (BesagandGreen, 1993), certainparameter transformations (Gelfand et al., 1995; Yu and Meng, 2011), and parameter expansion (Liu andWu,1999) canall beused to improve the convergenceof certain samplers.Byembedding an MH sampler within the Gibbs sampler and updating one parameter at a time (i.e. the well-known Metropolis-within-Gibbs sampler in the terminology of Gilks et al., 1995), the same strategies can be used to improve MH samplers.

Osteoporosis (OP) is a common metabolic bone disease with high incidence, and is recognized as a major public health problem worldwide. It is essential to clarify the pathogenesis of the disease for improving the diagnosis, prevention and treatment of OP. The aim of this study was to clarify the pivotal pathways and hub genes in OP using Gibbs sampling. The gene expression profile datasets were obtained from Gene Expression Omnibus (GEO) database. The pathways were enriched by Kyoto Encyclopedia of Genes and Genomes (KEGG) with genes intersection ≥5 based on gene expression profile data. Then, the acquired pathways were converted into Markov chains (MC). Gibbs sampling was conducted to obtain a new MC. In addition, the average probabilities of each pathway in two states containing human mesenchymal stem cells (hMSC) _middle-aged and hMSC_elderly were calculated through Markov chain Monte Carlo (MCMC) algorithm. Moreover, gene expression variation was taken into account to adjust the probability. Pivotal pathways were identified under adjusted posterior value >0.8. Then, Gibbs sampling was implemented to find hub genes from pathways. There were 280 pathways determined by the gene intersection ≥5. Gibbs sampling identified two disturbed pathways (pathways in cancer and influenza A) and two hub genes (cyclin A1 and WNT2) under the adjusted probability >0.8. Gene expression analysis showed that all the disturbed pathways and hub genes had increased expression levels in hMSC_middle-aged samples compared with hMSC_elderly samples. We identified two pivotal pathways and two hub genes in OP using Gibbs sampling. The results contribute to the understanding of underlying pathogenesis and could be considered as potential biomarkers for the therapy of OP.

This article provides an introduction to Markov chain Monte Carlo methods in statistical inference. Over the past twelve years or so, these have revolutionized what can be achieved computationally, especially in the Bayesian paradigm. Markov chain Monte Carlo has exactly the same goals as ordinary Monte Carlo and both are intended to exploit the fact that one can learn about a complex probability distribution if one can sample from it. Although the ordinary version can only rarely be implemented, it is convenient initially to presume otherwise and to focus on the rationale of the sampling approach, rather than computational details. The article then moves on to describe implementation via Markov chains, especially the Hastings algorithm, including the Metropolis method and the Gibbs sampler as special cases. Hidden Markov models and the autologistic distribution receive some emphasis, with the noisy binary channel used in some toy examples. A brief description of perfect simulation is also given. The account concludes with some discussion.Key wordsAutologistic distributionBayesian computationGibbs samplerHastings algorithmHidden Markov modelsImportance samplingIsing modelMarkov chain Monte CarloMarkov random fieldsMaximum likelihood estimationMetropolis methodNoisy binary channelPerfect simulationReversibilitySimulated annealing

Parameter estimation and uncertainty analysis for a watershed model

The epidemic of COVID-19 has been effectively controlled in China, which provides an empirical basis for intervention analysis. An extended SIR transmission model with heterogeneous force of infectiousness is presented including three fractals sub-models of Wuhan, Hubei and China. The model is built upon the data from January 20 to March 23, 2020. The Markov chain Monte Carlo (MCMC) Gibbs sampling and machine learning of Gaussian process regression are applied to solve the parameters estimation and the sensitivity analysis is used in various scenarios. The model has a breakthrough in parameter setting and accuracy optimization. It is found that contact reduction is the most effective core lever to control the epidemic, with 3.5 times or more sensitivity as much as other levers. The asymmetric marginal effect in strong and weak scenarios proves that Chinese controlling strategy is close to the Pareto Optimality. The intervention effectiveness of COVID-19 in China is evaluated by simulation analysis. The result provides valuable reference for China and other countries to judge the epidemic risk, control the intervention pace, and formulate prevention strategies in the future.

This paper tackles the estimation of parameters of linear mixed random effect one–classification model by Bayesian technique which includes Gibbs sampling. Gibbs sampling is a special case of Monte Carlo Method which uses Markov Chain and so called MCMC (Markov Chain Monte Carlo). This MCMC method depends on partition of difficult and compound models into simple ones which can be manipulated and easily analyzed, specially for the posterior distribution which are not easy to find their final formulae. In this research the mixed random effect linear one–classification model is proposed on a population of 15 treatments including 15 types of cotton plant. A random sample of 5 types is taken and using the analysis of variance method to test the hypothesis that all the 15 types have equal effect and the estimation of the parameters is obtained Gibbs sampling is also used in order to estimate the parameters and then testing the hypothesis of equal effects of treatments. The results obtained in both ANOVA and Gibbs sampling are nearly the same and encouraging. All algorithms are programmed in this research using WinBUGS program.

This article is a review article on the optimal Markov chain Monte Carlo (MCMC) sampling. The focus is on homogeneous Markov chains. This article first reviews the problem of finding the optimal transition matrix, which is defined to minimize the asymptotic variance of MCMC estimators. The article later reviews the locally optimal sampler (LOS), an MCMC sampling that performs local updates based on the optimal transition matrix. We conducted a simulation study to compare the LOS with the Metropolis–Hastings and the Gibbs Sampler. The LOS was shown to provide an improved rate of convergence over these two most popular sampling schemes. The implementation of the LOS requires only minor modifications in existing Gibbs sampling code. WIREs Comput Stat 2013. doi: 10.1002/wics.1265This article is categorized under: Statistical and Graphical Methods of Data Analysis > Markov Chain Monte Carlo (MCMC)

The Markov Chain Monte Carlo (MCMC) methods have become indispensable tools in modern statistical computation, enabling researchers to approximate complex probability distributions that are otherwise intractable. This paper focus on MCMC in Statistics and Probability area which is used to draw samples from a probability distribution. In order to introduce this algorithm in a relatively light and straightforward way, this paper breaks the content into two parts: Markov Chain and MCMC, and brings in stochastic process, Markov property, Ordinary Monte Carlo, and Monte Carlo Integration in succession. On this foundation, this paper formally introduces two algorithms of MCMC that are Metropolis-Hasting Algorithm and Gibbs Sampling, the most famous and significant algorithms of MCMC. With this approach, this paper concludes both the advantages and disadvantages of these two algorithms, laying the foundation of the further study and the deeper utilizations of MCMC and, hopefully, contributing to providing a certain degree of assistance for scholars.

In this study we compared the efficiency of frequentist approach with Bayesian approach by carrying out extreme value analysis of Annual Maximum Daily Rainfall (AMDR). For frequentist frequency analysis of AMDR, we used the data of one station i.e. Lahore in Punjab province, Pakistan while for Bayesian analysis we used the data of three other neighboring stations as prior information. During frequentist approach, Generalized Extreme Value (GEV) was found to be a best-fit distribution on the data. In frequentist method, the parameters of GEV distribution were estimated using Maximum Likelihood Estimation (MLE), while in the Bayesian framework the Markov Chain Monte Carlo (MCMC) simulation technique along with Metropolis-Hasting algorithm and Gibbs sampler were implemented. Findings of this study indicate that despite the asymptotic requirements of the MLE, its performance can be enhanced by adopting the Bayesian MCMC paradigm. In order to acquire the posterior densities of GEV parameters, non-informative and informative priors based on the historical data of surrounding weather stations were employed. The result of Bayesian MCMC might be affected by the choice of priors. In addition, the performance of the parameters estimation methods was verified by employing several robustness measures. Robustness measures results proved that the Bayesian MCMC method performed better than MLE in estimating GEV parameters and future return levels. Therefore, the findings of these analyses could be helpful in adopting proper flood protection measures and designing infrastructures of culverts, buildings, bridges and dams in the region.

Markov chainMonteCarlo (MCMC)methodshavebeenaround for almost as longasMonte Carlo techniques, even though their impact on statistics was not truly felt until the very early 1990s, except in the specialized fields of spatial statistics and image analysis, where those methods appeared earlier. The emergence of Markov based techniques in physics is a story that remains untold within this survey (see Landau and Binder, 2005). Also, we will not enter into a description of MCMC techniques, unless they have some historical link, as the remainder of this volume covers the technical aspects.Acomprehensive treatment with further references can also be found in Robert and Casella (2004). We will distinguish between the introduction of Metropolis-Hastings based algorithmsand those related to Gibbs sampling, since they each stem from radically different origins, even though their mathematical justification via Markov chain theory is the same. Tracing the development of Monte Carlo methods, we will also briefly mention what we might call the “second-generation MCMC revolution.” Starting in the mid to late 1990s, this includes the development of particle filters, reversible jump and perfect sampling, and concludes with more current work on population or sequential Monte Carlo and regeneration and the computing of “honest” standard errors. As mentioned above, the realization that Markov chains could be used in a wide varietyof situations only came (tomainstream statisticians)withGelfand and Smith (1990), despite earlier publications in the statistical literature such as Hastings (1970), Geman and Geman (1984), and Tanner and Wong (1987). Several reasons can be advanced: lack of computing machinery (think of the computers of 1970!), or background onMarkov chains, or hesitation to trust in the practicality of themethod. It thus required visionary researchers like Gelfand and Smith to convince the community, supported by papers that demonstrated, through a series of applications, that the method was easy to understand, easy to implement and practical (Gelfand et al., 1990, 1992; Smith and Gelfand, 1992; Wakefield et al., 1994). The rapid emergence of the dedicated BUGS (Bayesian inference using Gibbs sampling) software as early as 1991, when a paper on BUGS was presented at the Valencia meeting, was another compelling argument for adopting, at large, MCMC algorithms.∗Monte Carlo methods were born in Los Alamos, New Mexico, during World War II, eventually resulting in theMetropolis algorithm in the early 1950s. WhileMonte Carlo methods were in use by that time, MCMC was brought closer to statistical practicality by the work of Hastings in the 1970s. What can be reasonably seen as the first MCMC algorithm is what we now call theMetropolis algorithm, published by Metropolis et al. (1953). It emanates from the same group of scientists who produced the Monte Carlo method, namely the research scientists of Los Alamos, mostly physicists working on mathematical physics and the atomic bomb. MCMC algorithms therefore date back to the same time as the development of regular(MC only) Monte Carlo methods, which are usually traced to Ulam and von Neumann in the late 1940s. StanislawUlam associates the original ideawith an intractable combinatorial computation he attempted in 1946 (calculating the probability of winning at the solitaire card game). This ideawas enthusiastically adopted by John vonNeumann for implementationwith direct applications to neutron diffusion, the name “Monte Carlo” being suggested by Nicholas Metropolis. Eckhardt (1987) describes these early Monte Carlo developments, and Hitchcock (2003) gives a brief history of the Metropolis algorithm. These occurrences very closely coincide with the appearance of the very first general-purpose digital computer, the ENIAC,which came to life in February 1946, after three years of construction. The Monte Carlo method was set up by von Neumann, who was using it on thermonuclear and fission problems as early as 1947. That same year, Ulam and vonNeumann invented inversion andaccept-reject techniques (also recounted inEckhardt, 1987) to simulate from nonuniform distributions. Without computers, a rudimentary version invented by Fermi in the 1930s went unrecognized (Metropolis, 1987). Note also that, as early as 1949, a symposiumonMonteCarlowas supported byRand, theNational Bureau of Standards, and theOakRidge laboratory and thatMetropolis andUlam (1949) published the very first paper about the Monte Carlo method.

Markov chain Monte Carlo (MCMC) is a statistical innovation that allows researchers to fit far more com- plex models to data than is feasible using conventional methods. Despite its widespread use in a variety of scien- tific fields, MCMC appears to be underutilized in wildlife applications. This may be due to a misconception that MCMC requires the adoption of a subjective Bayesian analysis, or perhaps simply to its lack of familiarity among wildlife researchers. We introduce the basic ideas of MCMC and software BUGS (Bayesian inference using Gibbs sampling), stressing that a simple and satisfactory intuition for MCMC does not require extraordinary mathemat- ical sophistication. We illustrate the use of MCMC with an analysis of the association between latent factors gov- erning individual heterogeneity in breeding and survival rates of kittiwakes (Rissa tridactyla). We conclude with a discussion of the importance of individual heterogeneity for understanding population dynamics and designing management plans.

Summary1. A Bayesian analysis of site‐occupancy data containing covariates of species occurrence and species detection probabilities is usually completed using Markov chain Monte Carlo methods in conjunction with software programs that can implement those methods for any statistical model, not just site‐occupancy models. Although these software programs are quite flexible, considerable experience is often required to specify a model and to initialize the Markov chain so that summaries of the posterior distribution can be estimated efficiently and accurately.2. As an alternative to these programs, we develop a Gibbs sampler for Bayesian analysis of site‐occupancy data that include covariates of species occurrence and species detection probabilities. This Gibbs sampler is based on a class of site‐occupancy models in which probabilities of species occurrence and detection are specified as probit‐regression functions of site‐ and survey‐specific covariate measurements.3. To illustrate the Gibbs sampler, we analyse site‐occupancy data of the blue hawker,Aeshna cyanea(Odonata, Aeshnidae), a common dragonfly species in Switzerland. Our analysis includes a comparison of results based on Bayesian and classical (non‐Bayesian) methods of inference. We also provide code (based on the R software program) for conducting Bayesian and classical analyses of site‐occupancy data.

PurposeThe application of statistical inversion theory provides a powerful approach for solving estimation problems including the ability for uncertainty quantification (UQ) by means of Markov chain Monte Carlo (MCMC) methods and Monte Carlo integration. This paper aims to analyze the application of a state reduction technique within different MCMC techniques to improve the computational efficiency and the tuning process of these algorithms.Design/methodology/approachA reduced state representation is constructed from a general prior distribution. For sampling the Metropolis Hastings (MH) Algorithm and the Gibbs sampler are used. Efficient proposal generation techniques and techniques for conditional sampling are proposed and evaluated for an exemplary inverse problem.FindingsFor the MH-algorithm, high acceptance rates can be obtained with a simple proposal kernel. For the Gibbs sampler, an efficient technique for conditional sampling was found. The state reduction scheme stabilizes the ill-posed inverse problem, allowing a solution without a dedicated prior distribution. The state reduction is suitable to represent general material distributions.Practical implicationsThe state reduction scheme and the MCMC techniques can be applied in different imaging problems. The stabilizing nature of the state reduction improves the solution of ill-posed problems. The tuning of the MCMC methods is simplified.Originality/valueThe paper presents a method to improve the solution process of inverse problems within the Bayesian framework. The stabilization of the inverse problem due to the state reduction improves the solution. The approach simplifies the tuning of MCMC methods.

When a Markov Chain Monte Carlo (MCMC) method is applied to solve signal-processing problems, it is commonly implemented using Gibbs sampler. The implementation of Gibbs sampler requires the availability of full conditional probability density functions (pdfs) of all the parameters of interest of a problem. For some problems, however, the full conditional pdfs of all the parameters of interest are not readily available. In such cases, Metropolis-Hastings method can be incorporated within a Gibbs sampler to draw samples from the parameters whose full conditional pdf cannot be analytically determined. This paper demonstrates the application of such an algorithm, known as Metropolis-Hastings-within Gibbs, by considering the problem of joint data detection and channel estimation of a single-hop relay-based communication system. By formulating the signal model of the transmission process in alternative ways, we develop two algorithms for the problem. Moreover, simulation results of the two algorithms are provided to illustrate their effectiveness.