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Gareth O. Roberts - One of the best experts on this subject based on the ideXlab platform.
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stability of the Gibbs Sampler for bayesian hierarchical models
arXiv: Methodology, 2007Co-Authors: Omiros Papaspiliopoulos, Gareth O. RobertsAbstract:We characterise the convergence of the Gibbs Sampler which samples from the joint posterior distribution of parameters and missing data in hierarchical linear models with arbitrary symmetric error distributions. We show that the convergence can be uniform, geometric or sub-geometric depending on the relative tail behaviour of the error distributions, and on the parametrisation chosen. Our theory is applied to characterise the convergence of the Gibbs Sampler on latent Gaussian process models. We indicate how the theoretical framework we introduce will be useful in analyzing more complex models.
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Approximate Predetermined Convergence Properties of the Gibbs Sampler
Journal of Computational and Graphical Statistics, 2001Co-Authors: Gareth O. Roberts, Sujit K. SahuAbstract:This article aims to provide a method for approximately predetermining convergence properties of the Gibbs Sampler. This is to be done by first finding an approximate rate of convergence for a normal approximation of the target distribution. The rates of convergence for different implementation strategies of the Gibbs Sampler are compared to find the best one. In general, the limiting convergence properties of the Gibbs Sampler on a sequence of target distributions (approaching a limit) are not the same as the convergence properties of the Gibbs Sampler on the limiting target distribution. Theoretical results are given in this article to justify that under conditions, the convergence properties of the Gibbs Sampler can be approximated as well. A number of practical examples are given for illustration.
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on convergence of the em algorithmand the Gibbs Sampler
Statistics and Computing, 1999Co-Authors: Sujit K. Sahu, Gareth O. RobertsAbstract:In this article we investigate the relationship between the EM algorithm and the Gibbs Sampler. We show that the approximate rate of convergence of the Gibbs Sampler by Gaussian approximation is equal to that of the corresponding EM-type algorithm. This helps in implementing either of the algorithms as improvement strategies for one algorithm can be directly transported to the other. In particular, by running the EM algorithm we know approximately how many iterations are needed for convergence of the Gibbs Sampler. We also obtain a result that under certain conditions, the EM algorithm used for finding the maximum likelihood estimates can be slower to converge than the corresponding Gibbs Sampler for Bayesian inference. We illustrate our results in a number of realistic examples all based on the generalized linear mixed models.
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on the geometric convergence of the Gibbs Sampler
Journal of the royal statistical society series b-methodological, 1994Co-Authors: Gareth O. Roberts, Nicholas G PolsonAbstract:SUMMARY The rate of convergence of the Gibbs Sampler is discussed. The Gibbs Sampler is a Monte Carlo simulation method with extensive application to computational issues in the Bayesian paradigm. Conditions for the geometric rate of convergence of the algorithm for discrete and continuous parameter spaces are derived, and an illustrative exponential family example is given. This paper investigates conditions under which the Gibbs Sampler (Gelfand and Smith, 1990; Tanner and Wong, 1987; Geman and Geman, 1984) converges at a geometric rate. The main results appear in Sections 2 and 3, where geometric convergence results are established, with respect to total variation and supremum norms under fairly natural conditions on the underlying distribution. For ease of exposition, we shall concentrate on the two most commonly encountered situations, where the state space is finite or continuous. All our results will establish uniform convergence, a strong form of geometric convergence, under appropriate regularity conditions. Uniform convergence is a useful property in its own right but also happens to be a sufficient condition for certain ergodic central limit theorems. Such results are important for estimation in Markov chain simulation but will not be considered in detail here. Our approach is to apply the theory of Markov chains to the specific Gibbs Sampler case. In the finite state space case, uniform ergodicity is automatic. However, the situation is more complicated for continuous state spaces where even well-behaved underlying distributions can give rise to Markov chains which converge slowly, or have unbounded kernels. We give two results in this context, corollary 2 and corollary 3 which establish uniform convergence under different sets of conditions on the underlying density. Finally we apply these results to an example of a Bayesian hierarchical model where regularity conditions for geometric convergence are naturally satisfied. Here the hierarchical structure of the model is crucial in permitting application of corollary 3.
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simple conditions for the convergence of the Gibbs Sampler and metropolis hastings algorithms
Stochastic Processes and their Applications, 1994Co-Authors: Gareth O. Roberts, A F M SmithAbstract:Markov chain Monte Carlo (MCMC) simulation methods are being used increasingly in statistical computation to explore and estimate features of likelihood surfaces and Bayesian posterior distributions. This paper presents simple conditions which ensure the convergence of two widely used versions of MCMC, the Gibbs Sampler and Metropolis-Hastings algorithms.
C. Mandal - One of the best experts on this subject based on the ideXlab platform.
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A fast Gibbs Sampler for synthesizing constrained fractals
IEEE Transactions on Visualization and Computer Graphics, 1997Co-Authors: B.c. Vemuri, C. MandalAbstract:It is well known that the spatial frequency spectrum of membrane and thin plate splines exhibit self-affine characteristics and, hence, behave as fractals. This behavior was exploited in generating the constrained fractal surfaces, which were generated by using a Gibbs Sampler algorithm in the work of Szeliski and Terzopoulos (1989). The algorithm involves locally perturbing a constrained spline surface with white noise until the spline surface reaches an equilibrium state. We introduce a fast generalized Gibbs Sampler that combines two novel techniques, namely, a preconditioning technique in a wavelet basis for constraining the splines and a perturbation scheme in which, unlike the traditional Gibbs Sampler, all sites (surface nodes) that do not share a common neighbor are updated simultaneously. In addition, we demonstrate the capability to generate arbitrary order fractal surfaces without resorting to blending techniques. Using this fast Gibbs Sampler algorithm, we demonstrate the synthesis of realistic terrain models from sparse elevation data.
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IEEE Visualization - A fast Gibbs Sampler for synthesizing constrained fractals
Proceedings of Seventh Annual IEEE Visualization '96, 1996Co-Authors: C. Vemuri, C. MandalAbstract:It is well known that the spatial frequency spectra of membrane and thin-plate splines exhibit self-affine characteristics and hence behave as fractals. This behavior was exploited in generating the constrained fractal surfaces in the work of Szeliski and Terzopoulos (1989), which were generated by using a Gibbs Sampler algorithm. The algorithm involves locally perturbing a constrained spline surface with white noise until the spline surface reaches an equilibrium state. In this paper, we introduce a very fast generalized Gibbs Sampler that combines two novel techniques, namely a preconditioning technique in a wavelet basis for constraining the splines and a perturbation scheme in which, unlike the traditional Gibbs Sampler, all sites (surface nodes) that do not share a common neighbor are updated simultaneously. In addition, we demonstrate the capability to generate arbitrary-order fractal surfaces without resorting to blending techniques. Using this fast Gibbs Sampler algorithm, we demonstrate the synthesis of realistic terrain models from sparse elevation data.
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A fast Gibbs Sampler for synthesizing constrained fractals
Proceedings of Seventh Annual IEEE Visualization '96, 1996Co-Authors: B.c. Vemuri, C. MandalAbstract:It is well known that the spatial frequency spectra of membrane and thin-plate splines exhibit self-affine characteristics and hence behave as fractals. This behavior was exploited in generating the constrained fractal surfaces in the work of Szeliski and Terzopoulos (1989), which were generated by using a Gibbs Sampler algorithm. The algorithm involves locally perturbing a constrained spline surface with white noise until the spline surface reaches an equilibrium state. In this paper, we introduce a very fast generalized Gibbs Sampler that combines two novel techniques, namely a preconditioning technique in a wavelet basis for constraining the splines and a perturbation scheme in which, unlike the traditional Gibbs Sampler, all sites (surface nodes) that do not share a common neighbor are updated simultaneously. In addition, we demonstrate the capability to generate arbitrary-order fractal surfaces without resorting to blending techniques. Using this fast Gibbs Sampler algorithm, we demonstrate the synthesis of realistic terrain models from sparse elevation data.
A F M Smith - One of the best experts on this subject based on the ideXlab platform.
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simple conditions for the convergence of the Gibbs Sampler and metropolis hastings algorithms
Stochastic Processes and their Applications, 1994Co-Authors: Gareth O. Roberts, A F M SmithAbstract:Markov chain Monte Carlo (MCMC) simulation methods are being used increasingly in statistical computation to explore and estimate features of likelihood surfaces and Bayesian posterior distributions. This paper presents simple conditions which ensure the convergence of two widely used versions of MCMC, the Gibbs Sampler and Metropolis-Hastings algorithms.
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bayesian computation via the Gibbs Sampler and related markov chain monte carlo methods
Journal of the royal statistical society series b-methodological, 1993Co-Authors: A F M Smith, Gareth O. RobertsAbstract:The use of the Gibbs Sampler for Bayesian computation is reviewed and illustrated in the context of some canonical examples. Other Markov chain Monte Carlo simulation methods are also briefly described, and comments are made on the advantages of sample-based approaches for Bayesian inference summaries
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bayesian computations in survival models via the Gibbs Sampler
1992Co-Authors: A F M SmithAbstract:Survival models used in biomedical and reliability contexts typically involve data censoring, and may also involve constraints in the form of ordered parameters. In addition, inferential interest often focuses on non-linear functions of natural model parameters. From a Bayesian statistical analysis perspective, these features combine to create difficult computational problems by seeming to require (multi-dimensional) numerical integrals over awkwardly defined regions. This paper illustrates how these apparent difficulties can be overcome, in both parametric and nonparametric settings, by the Gibbs Sampler approach to Bayesian computation.
Daniel Taylor Rodriguez - One of the best experts on this subject based on the ideXlab platform.
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a Gibbs Sampler for bayesian analysis of site occupancy data
Methods in Ecology and Evolution, 2012Co-Authors: Robert M Dorazio, Daniel Taylor RodriguezAbstract:Summary 1. 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.
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A Gibbs Sampler for Bayesian analysis of site‐occupancy data
Methods in Ecology and Evolution, 2012Co-Authors: Robert M Dorazio, Daniel Taylor RodriguezAbstract:Summary 1. 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.
B.c. Vemuri - One of the best experts on this subject based on the ideXlab platform.
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A fast Gibbs Sampler for synthesizing constrained fractals
IEEE Transactions on Visualization and Computer Graphics, 1997Co-Authors: B.c. Vemuri, C. MandalAbstract:It is well known that the spatial frequency spectrum of membrane and thin plate splines exhibit self-affine characteristics and, hence, behave as fractals. This behavior was exploited in generating the constrained fractal surfaces, which were generated by using a Gibbs Sampler algorithm in the work of Szeliski and Terzopoulos (1989). The algorithm involves locally perturbing a constrained spline surface with white noise until the spline surface reaches an equilibrium state. We introduce a fast generalized Gibbs Sampler that combines two novel techniques, namely, a preconditioning technique in a wavelet basis for constraining the splines and a perturbation scheme in which, unlike the traditional Gibbs Sampler, all sites (surface nodes) that do not share a common neighbor are updated simultaneously. In addition, we demonstrate the capability to generate arbitrary order fractal surfaces without resorting to blending techniques. Using this fast Gibbs Sampler algorithm, we demonstrate the synthesis of realistic terrain models from sparse elevation data.
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A fast Gibbs Sampler for synthesizing constrained fractals
Proceedings of Seventh Annual IEEE Visualization '96, 1996Co-Authors: B.c. Vemuri, C. MandalAbstract:It is well known that the spatial frequency spectra of membrane and thin-plate splines exhibit self-affine characteristics and hence behave as fractals. This behavior was exploited in generating the constrained fractal surfaces in the work of Szeliski and Terzopoulos (1989), which were generated by using a Gibbs Sampler algorithm. The algorithm involves locally perturbing a constrained spline surface with white noise until the spline surface reaches an equilibrium state. In this paper, we introduce a very fast generalized Gibbs Sampler that combines two novel techniques, namely a preconditioning technique in a wavelet basis for constraining the splines and a perturbation scheme in which, unlike the traditional Gibbs Sampler, all sites (surface nodes) that do not share a common neighbor are updated simultaneously. In addition, we demonstrate the capability to generate arbitrary-order fractal surfaces without resorting to blending techniques. Using this fast Gibbs Sampler algorithm, we demonstrate the synthesis of realistic terrain models from sparse elevation data.