Rosenthal, J. S. (1991). Rates of Convergence for Gibbs Sampling for Variance Component Models. Tech. Rep. 9322, Department of Statistics, University of Toronto.  Home/Search   Document Details and Download   Summary   Related Articles   Check

..... SpringerVerlag, London. 10] Nummelin, E. 1984) General Irreducible Markov Chains and Non negative Operators. Cambridge, London. 11] Roberts, G. O. and Polson, N. G. 1994) On the geometric convergence of the Gibbs sampler. J. R. Statist. Soc. B. 56 377 384. 12] Roberts, G. O. and Rosenthal, J. S. 1997) Geometric ergodicity and hybrid Markov chains. Elect. Comm. in Probab. 2 13 25. 13] Roberts, G. O. and Rosenthal, J. S. 1998) Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion) To appear in Canad. J. of Statist. 14] Roberts, G. O. and ....

....[11] Roberts, G. O. and Polson, N. G. 1994) On the geometric convergence of the Gibbs sampler. J. R. Statist. Soc. B. 56 377 384. 12] Roberts, G. O. and Rosenthal, J. S. 1997) Geometric ergodicity and hybrid Markov chains. Elect. Comm. in Probab. 2 13 25. 13] Roberts, G. O. and Rosenthal, J. S. 1998) Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion) To appear in Canad. J. of Statist. 14] Roberts, G. O. and Sahu, S. K. 1997) Updating schemes, correlation structure, blocking and parameterization for the Gibbs sampler. J. R. Statist. ....

[Article contains additional citation context not shown here]

Rosenthal, J. S. (1995b). Rates of convergence for Gibbs sampling for variance component models, The Annals of Statistics 23: 740--761.

....will converge to the true posterior after O(log n) steps. Thus, the number of steps required to approach the true posterior does not grow too quickly with the amount of observed data. This suggests the feasibility of running this iterative process when given a large but nite amount of data. In [R], similar results are obtained for a more complicated model, namely the variance component models as discussed in [GS] The plan of this paper is as follows. In Section 2 we review the de nition of the Data Augmentation algorithm, and state the key lemma to be used in proving convergence ....

.... should be used in these cases, but rather that the convergence results obtained here may provide some insight into using Data Augmentation and Gibbs Sampler in more complicated examples, such as those considered in [GS] and [GHRS] We intend to consider some of those examples elsewhere [R]. Also, the methods used here may be applicable to many other Markov chain problems. The main tool used in proving the above result will be the following Upper Bound Lemma , inspired by the discussion on page 151 of [A] It is closely related to the notions of Doeblin and Harris recurrence (see ....

[Article contains additional citation context not shown here]

J.S. Rosenthal (1991), Rates of convergence for Gibbs sampling for variance component models, Tech. Rep., Dept. of Mathematics, Harvard University.

.... question has been motivated by such diverse areas as card shuffling ( How many times do you have to shuffle a deck of cards to make it random ; see [D] for background) and stochastic algorithms ( How long do you have to run the algorithm until the answers are satisfactory ; see e.g. GS] and [R]) In each case, it is desired to know how long a Markov chain should be run until it has converged to the desired stationary distribution. The study of non asymptotic convergence rates often yields interesting results. The best known of these is the cut off phenomonon of Diaconis and Shashahani ....

J.S. Rosenthal (1991), Rates of convergence for Gibbs sampling for variance component models, Technical Report, University of Minnesota. School of Mathematics University of Minnesota Minneapolis, MN 55455, U.S.A. 32

....See [D] for background, examples, and references. For random walks on compact Lie groups, there has been some recent progress; see [R1] For more general Markov chains, the notion of Harris recurrence (see [A] AN] N] has proven useful in obtaining rates of convergence (see e.g. T] R2] [R3]) Finite state space Markov chains remain the simplest case to study, because their convergence can be analyzed directly in terms of the finite spectrum of their transition kernel; see e.g. DS] In this paper, we identify a class of Markov chains, which we call pseudo finite , which are ....

J.S. Rosenthal (1991), Rates of convergence for Gibbs sampling for variance component models, preprint.

No context found.

Rosenthal, J.S. (1995a). Rates of convergence for Gibbs sampling for variance components models. Ann. Statist., 23, 740-761.

....Metropolis Hastings algorithm, are very commonly used in Bayesian statistics for sampling from complicated, high dimensional posterior distributions. A continuing source of uncertainty is how long such a sampler must be run in order to converge approximately to its target stationary distribution. Rosenthal (1995b) presents a method to compute rigorous theoretical upper bounds on the number of iterations required to achieve a specified degree of convergence in total variation distance by verifying drift and minorization conditions. We propose the use of auxiliary simulations to estimate the numerical ....

.... distribution That is, how large should the burn in time be Rigorous theoretical upper bounds on burn in times for these algorithms have recently been proposed (see e.g. Frieze, Kannan, and Polson, 1993; Frigessi, Hwang, Sheu, and di Stefano, 1993; Ingrassia, 1994; Meyn and Tweedie, 1994; Rosenthal, 1995b; Baxendale, 1994) However, they have suffered from the difficulties of precise analysis of complicated models, and have largely tended to concentrate on relatively simple problems, and or to provide impractically large upper bounds. Consequently, most applied users of MCMC techniques have used ....

[Article contains additional citation context not shown here]

Rosenthal, J.S. (1995a), "Rates of Convergence for Gibbs Sampling for Variance Components Models." Annals of Statistics 23 (1995), 740--761.

....will converge to the true posterior after O(log n) steps. Thus, the number of steps required to approach the true posterior does not grow too quickly with the amount of observed data. This suggests the feasibility of running this iterative process when given a large but finite amount of data. In [R], similar results are obtained for a more complicated model, namely the variance component models as discussed in [GS] The plan of this paper is as follows. In Section 2 we review the definition of the Data Augmentation algorithm, and state the key lemma to be used in proving convergence results. ....

.... should be used in these cases, but rather that the convergence results obtained here may provide some insight into using Data Augmentation and Gibbs Sampler in more complicated examples, such as those considered in [GS] and [GHRS] We intend to consider some of those examples elsewhere [R]. Also, the methods used here may be applicable to many other Markov chain problems. The main tool used in proving the above result will be the following Upper Bound Lemma , inspired by the discussion on page 151 of [A] It is closely related to the notions of Doeblin and Harris recurrence (see ....

[Article contains additional citation context not shown here]

J.S. Rosenthal (1991), Rates of convergence for Gibbs sampling for variance component models, Tech. Rep., Dept. of Mathematics, Harvard University.

....See [D] for background, examples, and references. For random walks on compact Lie groups, there has been some recent progress; see [R1] For more general Markov chains, the notion of Harris recurrence (see [A] AN] N] has proven useful in obtaining rates of convergence (see e.g. T] R2] [R3]) Finite state space Markov chains remain the simplest case to study, because their convergence can be analyzed directly in terms of the finite spectrum of their transition kernel; see e.g. DS] In this paper, we identify a class of Markov chains, which we call pseudo finite , which are ....

J.S. Rosenthal (1991), Rates of convergence for Gibbs sampling for variance component models, preprint.

.... question has been motivated by such diverse areas as card shuffling ( How many times do you have to shuffle a deck of cards to make it random ; see [D] for background) and stochastic algorithms ( How long do you have to run the algorithm until the answers are satisfactory ; see e.g. GS] and [R]) In each case, it is desired to know how long a Markov chain should be run until it has converged to the desired stationary distribution. The study of non asymptotic convergence rates often yields interesting results. The best known of these is the cut off phenomonon of Diaconis and Shashahani ....

J.S. Rosenthal (1991), Rates of convergence for Gibbs sampling for variance component models, Technical Report, University of Minnesota. School of Mathematics University of Minnesota Minneapolis, MN 55455, U.S.A.

No context found.

Rosenthal, J.S. (1995a). Rates of convergence for Gibbs sampling for variance components models. Ann. Statist., 23, 740-761.

No context found.

Rosenthal, J. S. (1991). Rates of Convergence for Gibbs Sampling for Variance Component Models. Tech. Rep. 9322, Department of Statistics, University of Toronto.

No context found.

Rosenthal, J.S. (1995a). "Rates of convergence for Gibbs sampling for variance components models". Annals of Statistics (to appear).

No context found.

Rosenthal, J.S. (1995c) "Rates of Convergence for Gibbs Sampling for Variance Component Models," to appear in Annals of Statistics.

No context found.

Rosenthal, J. (1991), "Rates of Convergence for Gibbs Sampling for Variance Component Models," Technical Report, Harvard University, Dept. of Mathematics.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC