A Markov chain Monte Carlo strategy for sampling from the joint posterior distribution of pedigrees and population parameters under a Fisher–Wright model with partial selfing

2007 ◽  
Vol 72 (3) ◽  
pp. 436-458 ◽  
Author(s):  
Ian J. Wilson ◽  
Kevin J. Dawson
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Posterior Distribution ◽  
Population Parameters ◽  
Joint Posterior Distribution

scholarly journals A Comparison of Penalized Maximum Likelihood Estimation and Markov Chain Monte Carlo Techniques for Estimating Confirmatory Factor Analysis Models with Small Sample Sizes

2020 ◽  
Author(s):  
Oliver Lüdtke ◽  
Esther Ulitzsch ◽  
Alexander Robitzsch
Keyword(s):  
Monte Carlo ◽  
Factor Analysis ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Maximum Likelihood ◽  
Posterior Distribution ◽  
Parameter Estimates ◽  
Penalized Maximum Likelihood ◽  
Confirmatory Factor ◽  
Joint Posterior Distribution

With small to modest sample sizes and complex models, maximum likelihood (ML) estimation of confirmatory factor analysis (CFA) models can show serious estimation problems such as nonconvergence or parameter estimates that are outside the admissible parameter space. In the present article, we discuss two Bayesian estimation methods for stabilizing parameter estimates of a CFA: Penalized maximum likelihood (PML) estimation and Markov Chain Monte Carlo (MCMC) methods. We clarify that these use different Bayesian point estimates from the joint posterior distribution—the mode (PML) of the joint posterior distribution, and the mean (EAP) or mode (MAP) of the marginal posterior distribution—and discuss under which conditions the two methods produce different results. In a simulation study, we show that the MCMC method clearly outperforms PML and that these performance gains can be explained by the fact that MCMC uses the EAP as a point estimate. We also argue that it is often advantageous to choose a parameterization in which the main parameters of interest are bounded and suggest the four-parameter beta distribution as a prior distribution for loadings and correlations. Using simulated data, we show that selecting weakly informative four-parameter beta priors can further stabilize parameter estimates, even in cases when the priors were mildly misspecified. Finally, we derive recommendations and propose directions for further research.

Markov Chain Monte Carlo

2015 ◽  
Author(s):  
N. Thompson Hobbs ◽  
Mevin B. Hooten
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Bayesian Analysis ◽  
Posterior Distribution ◽  
Mcmc Algorithm ◽  
Bayesian Analyses ◽  
Seminal Paper ◽  
Formal Treatment ◽  
High Level

This chapter explains how to implement Bayesian analyses using the Markov chain Monte Carlo (MCMC) algorithm, a set of methods for Bayesian analysis made popular by the seminal paper of Gelfand and Smith (1990). It begins with an explanation of MCMC with a heuristic, high-level treatment of the algorithm, describing its operation in simple terms with a minimum of formalism. In this first part, the chapter explains the algorithm so that all readers can gain an intuitive understanding of how to find the posterior distribution by sampling from it. Next, the chapter offers a somewhat more formal treatment of how MCMC is implemented mathematically. Finally, this chapter discusses implementation of Bayesian models via two routes—by using software and by writing one's own algorithm.

Efficient History Matching Using the Markov-Chain Monte Carlo Method by Means of the Transformed Adaptive Stochastic Collocation Method

SPE Journal ◽  
2019 ◽  
Vol 24 (04) ◽  
pp. 1468-1489 ◽  
Author(s):  
Qinzhuo Liao ◽  
Lingzao Zeng ◽  
Haibin Chang ◽  
Dongxiao Zhang
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Collocation Method ◽  
Posterior Distribution ◽  
History Matching ◽  
Stochastic Collocation ◽  
Mcmc Method ◽  
Stochastic Collocation Method ◽  
Surrogate System

Summary Bayesian inference provides a convenient framework for history matching and prediction. In this framework, prior knowledge, system nonlinearity, and measurement errors can be directly incorporated into the posterior distribution of the parameters. The Markov-chain Monte Carlo (MCMC) method is a powerful tool to generate samples from the posterior distribution. However, the MCMC method usually requires a large number of forward simulations. Hence, it can be a computationally intensive task, particularly when dealing with large-scale flow and transport models. To address this issue, we construct a surrogate system for the model outputs in the form of polynomials using the stochastic collocation method (SCM). In addition, we use interpolation with the nested sparse grids and adaptively take into account the different importance of parameters for high-dimensional problems. Furthermore, we introduce an additional transform process to improve the accuracy of the surrogate model in case of strong nonlinearities, such as a discontinuous or unsmooth relation between the input parameters and the output responses. Once the surrogate system is built, we can evaluate the likelihood with little computational cost. Numerical results demonstrate that the proposed method can efficiently estimate the posterior statistics of input parameters and provide accurate results for history matching and prediction of the observed data with a moderate number of parameters.

A Fast and Simple Algorithm for Bayesian Adaptive Testing

2019 ◽  
Vol 45 (1) ◽  
pp. 58-85
Author(s):  
Wim J. van der Linden ◽  
Hao Ren
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Simple Algorithm ◽  
Adaptive Testing ◽  
Monte Carlo Algorithm ◽  
Rapid Convergence ◽  
Point Estimates ◽  
Item Parameters ◽  
Joint Posterior Distribution

The Bayesian way of accounting for the effects of error in the ability and item parameters in adaptive testing is through the joint posterior distribution of all parameters. An optimized Markov chain Monte Carlo algorithm for adaptive testing is presented, which samples this distribution in real time to score the examinee’s ability and optimally select the items. Thanks to extremely rapid convergence of the Markov chain and simple posterior calculations, the algorithm is ready for use in real-world adaptive testing with running times fully comparable with algorithms that fix all parameters at point estimates during testing.

scholarly journals SAR IMAGE SEGMENTATION WITH UNKNOWN NUMBER OF CLASSES COMBINED VORONOI TESSELLATION AND RJMCMC ALGORITHM

2016 ◽  
Vol III-7 ◽  
pp. 119-124 ◽  
Author(s):  
Q. H. Zhao ◽  
Y. Li ◽  
Y. Wang
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Posterior Distribution ◽  
Voronoi Tessellation ◽  
Random Variable ◽  
Model Parameters ◽  
Sar Image ◽  
Sar Images ◽  
Number Of Classes

This paper presents a novel segmentation method for automatically determining the number of classes in Synthetic Aperture Radar (SAR) images by combining Voronoi tessellation and Reversible Jump Markov Chain Monte Carlo (RJMCMC) strategy. Instead of giving the number of classes <i>a priori</i>, it is considered as a random variable and subject to a Poisson distribution. Based on Voronoi tessellation, the image is divided into homogeneous polygons. By Bayesian paradigm, a posterior distribution which characterizes the segmentation and model parameters conditional on a given SAR image can be obtained up to a normalizing constant; Then, a Revisable Jump Markov Chain Monte Carlo(RJMCMC) algorithm involving six move types is designed to simulate the posterior distribution, the move types including: splitting or merging real classes, updating parameter vector, updating label field, moving positions of generating points, birth or death of generating points and birth or death of an empty class. Experimental results with real and simulated SAR images demonstrate that the proposed method can determine the number of classes automatically and segment homogeneous regions well.

scholarly journals Properties of Markov Chain Monte Carlo Performance across Many Empirical Alignments

2020 ◽  
Author(s):  
Sean M Harrington ◽  
Van Wishingrad ◽  
Robert C Thomson
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Posterior Distribution ◽  
Common Knowledge ◽  
Simulated Data ◽  
Mcmc Methods ◽  
Rate Variation ◽  
Data Sets ◽  
Mcmc Convergence

Abstract Nearly all current Bayesian phylogenetic applications rely on Markov chain Monte Carlo (MCMC) methods to approximate the posterior distribution for trees and other parameters of the model. These approximations are only reliable if Markov chains adequately converge and sample from the joint posterior distribution. Although several studies of phylogenetic MCMC convergence exist, these have focused on simulated data sets or select empirical examples. Therefore, much that is considered common knowledge about MCMC in empirical systems derives from a relatively small family of analyses under ideal conditions. To address this, we present an overview of commonly applied phylogenetic MCMC diagnostics and an assessment of patterns of these diagnostics across more than 18,000 empirical analyses. Many analyses appeared to perform well and failures in convergence were most likely to be detected using the average standard deviation of split frequencies, a diagnostic that compares topologies among independent chains. Different diagnostics yielded different information about failed convergence, demonstrating that multiple diagnostics must be employed to reliably detect problems. The number of taxa and average branch lengths in analyses have clear impacts on MCMC performance, with more taxa and shorter branches leading to more difficult convergence. We show that the usage of models that include both Γ-distributed among-site rate variation and a proportion of invariable sites is not broadly problematic for MCMC convergence but is also unnecessary. Changes to heating and the usage of model-averaged substitution models can both offer improved convergence in some cases, but neither are a panacea.

scholarly journals SAR IMAGE SEGMENTATION WITH UNKNOWN NUMBER OF CLASSES COMBINED VORONOI TESSELLATION AND RJMCMC ALGORITHM

2016 ◽  
Vol III-7 ◽  
pp. 119-124
Author(s):  
Q. H. Zhao ◽  
Y. Li ◽  
Y. Wang
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Posterior Distribution ◽  
Voronoi Tessellation ◽  
Random Variable ◽  
Model Parameters ◽  
Sar Image ◽  
Sar Images ◽  
Number Of Classes

This paper presents a novel segmentation method for automatically determining the number of classes in Synthetic Aperture Radar (SAR) images by combining Voronoi tessellation and Reversible Jump Markov Chain Monte Carlo (RJMCMC) strategy. Instead of giving the number of classes <i>a priori</i>, it is considered as a random variable and subject to a Poisson distribution. Based on Voronoi tessellation, the image is divided into homogeneous polygons. By Bayesian paradigm, a posterior distribution which characterizes the segmentation and model parameters conditional on a given SAR image can be obtained up to a normalizing constant; Then, a Revisable Jump Markov Chain Monte Carlo(RJMCMC) algorithm involving six move types is designed to simulate the posterior distribution, the move types including: splitting or merging real classes, updating parameter vector, updating label field, moving positions of generating points, birth or death of generating points and birth or death of an empty class. Experimental results with real and simulated SAR images demonstrate that the proposed method can determine the number of classes automatically and segment homogeneous regions well.

scholarly journals Diffusion Equation-Assisted Markov Chain Monte Carlo Methods for the Inverse Radiative Transfer Equation

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 291 ◽  
Author(s):  
Qin Li ◽  
Kit Newton
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Radiative Transfer ◽  
Diffusion Equation ◽  
Posterior Distribution ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Optical Tomography ◽  
Scattering Coefficient

Optical tomography is the process of reconstructing the optical properties of biological tissue using measurements of incoming and outgoing light intensity at the tissue boundary. Mathematically, light propagation is modeled by the radiative transfer equation (RTE), and optical tomography amounts to reconstructing the scattering coefficient in the RTE using the boundary measurements. In the strong scattering regime, the RTE is asymptotically equivalent to the diffusion equation (DE), and the inverse problem becomes reconstructing the diffusion coefficient using Dirichlet and Neumann data on the boundary. We study this problem in the Bayesian framework, meaning that we examine the posterior distribution of the scattering coefficient after the measurements have been taken. However, sampling from this distribution is computationally expensive, since to evaluate each Markov Chain Monte Carlo (MCMC) sample, one needs to run the RTE solvers multiple times. We therefore propose the DE-assisted two-level MCMC technique, in which bad samples are filtered out using DE solvers that are significantly cheaper than RTE solvers. This allows us to make sampling from the RTE posterior distribution computationally feasible.

Multimodal Markov chain Monte Carlo method for nonlinear petrophysical seismic inversion

Geophysics ◽  
2019 ◽  
Vol 84 (5) ◽  
pp. M1-M13 ◽  
Author(s):  
Leandro Passos de Figueiredo ◽  
Dario Grana ◽  
Mauro Roisenberg ◽  
Bruno B. Rodrigues
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Posterior Distribution ◽  
Seismic Inversion ◽  
Forward Model ◽  
Model Parameters ◽  
Petrophysical Properties ◽  
Mcmc Method ◽  
Data Set

One of the main objectives in the reservoir characterization is estimating the rock properties based on seismic measurements. We have developed a stochastic sampling method for the joint prediction of facies and petrophysical properties, assuming a nonparametric mixture prior distribution and a nonlinear forward model. The proposed methodology is based on a Markov chain Monte Carlo (MCMC) method specifically designed for multimodal distributions for nonlinear problems. The vector of model parameters includes the facies sequence along the seismic trace as well as the continuous petrophysical properties, such as porosity, mineral fractions, and fluid saturations. At each location, the distribution of petrophysical properties is assumed to be multimodal and nonparametric with as many modes as the number of facies; therefore, along the seismic trace, the distribution is multimodal with the number of modes being equal to the number of facies power the number of samples. Because of the nonlinear forward model, the large number of modes and as a consequence the large dimension of the model space, the analytical computation of the full posterior distribution is not feasible. We then numerically evaluate the posterior distribution by using an MCMC method in which we iteratively sample the facies, by moving from one mode to another, and the petrophysical properties, by sampling within the same mode. The method is extended to multiple seismic traces by applying a first-order Markov chain that accounts for the lateral continuity of the model properties. We first validate the method using a synthetic 2D reservoir model and then we apply the method to a real data set acquired in a carbonate field.

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