Above are papers related to the following paper: [Paper_ID] Paper #1131335. 	[Title] Estimating L 1 Error of Kernel Estimator: Monitoring Convergence of Markov Samplers [Abstract] In many Markov chain Monte Carlo problems, the target density function is known up to a normalization constant. In this paper, we take advantage of this knowledge to facilitate the convergence diagnostic of a Markov sampler by estimating the L 1 error of a kernel estimator. Firstly, we propose an estimator of the normalization constant which is shown to be asymptotically normal under mixing and moment conditions. Secondly, the L 1 error of the kernel estimator is estimated using the normalization constant estimator, and the ratio of the estimated L 1 error to the true L 1 error is shown to converge to 1 in probability under similar conditions. Thirdly, we propose a sequential plot of the estimated L 1 error as a tool to monitor the convergence of the Markov sampler. Finally, a 2-dimensional bimodal example is given to illustrate the proposal, fl Bin Yu is Assistant Professor, Department of Statistics, University of California, Berkeley, CA 94720-3860. Research supported in part by the Junior Faculty Research Grant from University of California at Berkeley, grants DAAL03-91-G-007 and DAAH04-94-G-0232 from the Army Research Office, and grant DMS-9322817 from the National Science Foundation. The author is very grateful to Professors Peter Bickel and Andrew Gelman for many helpful discussions and their comments on the draft. Special thanks are due to Mr. Sam Buttrey for his help on simulation, to Professor Per Mykland and Mr. Karl Broman for commenting on the draft, and to two anonymous and two Markov samplers are compared in the example using the proposed diagnostic plot. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #645897. 	[Title] A note on convergence rates of Gibbs sampling for nonparametric mixtures [Abstract] We consider a mixture model where the mixing distribution is random and is given a Dirichlet process prior. We describe the general structure of two Gibbs sampling algorithms that are useful for approximating Bayesian inferences in this problem. When the kernel f(x j ) of the mixture is bounded, we show that the Markov chains resulting from the Gibbs sampling are uniformly ergodic, and we provide an explicit rate bound. Unfortunately, the bound is not sharp in general; improving sensibly the bound seems however quite difficult. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #646286. 	[Title] Rate of Convergence of the Gibbs Sampler by Gaussian Approximation  SUMMARY [Abstract] In this article we approximate the rate of convergence of the Gibbs sampler by a normal approximation of the target distribution. Based on this approximation, we consider many implementational issues for the Gibbs sampler, e.g., updating strategy, parameterization and blocking. We give theoretical results to justify our approximation and illustrate our methods in a number of realistic examples. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #643221. 	[Title] A simulation approach to convergence rates for Markov chain Monte Carlo algorithms [Abstract] Markov chain Monte Carlo (MCMC) methods, including the Gibbs sampler and the 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 values needed in Rosenthal's theorem. Our simulation method makes it possible to compute quantitative convergence bounds for models for which the requisite analytical computations would be prohibitively difficult or impossible. On the other hand, although our method appears to perform well in our example problems, it can not provide the guarantees offered by analytical proof. Acknowledgements. We thank Brad Carlin for assistance and encouragement. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #643485. 	[Title] Geometric Ergodicity of Gibbs and Block Gibbs Samplers for a Hierarchical Random Effects Model [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1131301. 	[Title] A note on acceptance rate criteria for CLTs for Hastings-Metropolis algorithms [Abstract] This note considers positive recurrent Markov chains where the probability of remaining in the current state is arbitrarily close to 1. Specifically, conditions are given which ensure the non-existence of central limit theorems for ergodic averages of functionals of the chain. The results are motivated by applications for Metropolis-Hastings algorithms which are constructed in terms of a rejection probability, (where a rejection involves remaining at the current state). Two examples for commonly used algorithms are given, for the independence sampler and the Metropolis adjusted Langevin algorithm. The examples are rather specialised, although in both cases, the problems which arise are typical of problems commonly occurring for the particular algorithm being used. 0 I would like to thank Kerrie Mengersen Jeff Rosenthal and Richard Tweedie for useful conversations on the subject of this paper. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #643239. 	[Title] Analysis of the Gibbs sampler for a model related to James-Stein estimators [Abstract] Summary. We analyze a hierarchical Bayes model which is related to the usual empirical Bayes formulation of James-Stein estimators. We consider running a Gibbs sampler on this model. Using previous results about convergence rates of Markov chains, we provide rigorous, numerical, reasonable bounds on the running time of the Gibbs sampler, for a suitable range of prior distributions. We apply these results to baseball data from Efron and Morris (1975). For a different range of prior distributions, we prove that the Gibbs sampler will fail to converge, and use this information to prove that in this case the associated posterior distribution is non-normalizable. Acknowledgements. I am very grateful to Jun Liu for suggesting this project, and to Neal Madras for suggesting the use of the Submartingale Convergence Theorem herein. I thank Kate Cowles and Richard Tweedie for helpful conversations, and thank the referees for useful comments. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1131305. 	[Title] Two convergence properties of hybrid samplers [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1131180. 	[Title] EXACT BOUND FOR THE CONVERGENCE OF METROPOLIS CHAINS [Abstract] In this note, we present a calculation which gives us the exact bound for the convergence of independent Metropolis chains in a finite state space. Metropolis chain, convergence rate, Markov chain Monte Carlo Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1131312. 	[Title] Theoretical rates of convergence for Markov chain Monte Carlo [Abstract] We present a general method for proving rigorous, a priori bounds on the number of iterations required to achieve convergence of Markov chain Monte Carlo. We describe bounds for specific models of the Gibbs sampler, which have been obtained from the general method. We discuss possibilities for obtaining bounds more generally. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #16819. 	[Title] Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review [Abstract] A critical issue for users of Markov Chain Monte Carlo (MCMC) methods in applications is how to determine when it is safe to stop sampling and use the samples to estimate characteristics of the distribution of interest. Research into methods of computing theoretical convergence bounds holds promise for the future but currently has yielded relatively little that is of practical use in applied work. Consequently, most MCMC users address the convergence problem by applying diagnostic tools to the output produced by running their samplers. After giving a brief overview of the area, we provide an expository review of thirteen convergence diagnostics, describing the theoretical basis and practical implementation of each. We then compare their performance in two simple models and conclude that all the methods can fail to detect the sorts of convergence failure they were designed to identify. We thus recommend a combination of strategies aimed at evaluating and accelerating MCMC sampler convergence, including applying diagnostic procedures to a small number of parallel chains, monitoring autocorrelations and cross-correlations, and modifying parameterizations or sampling algorithms appropriately. We emphasize, however, that it is not possible to say with certainty that a finite sample from an MCMC algorithm is representative of an underlying stationary distribution. Mary Kathryn Cowles is Assistant Professor of Biostatistics, Harvard School of Public Health, Boston, MA 02115. Bradley P. Carlin is Associate Professor, Division of Biostatistics, School of Public Health, University of Minnesota, Minneapolis, MN 55455. Much of the work was done while the first author was a graduate student in the Divison of Biostatistics at the University of Minnesota and then Assistant Professor, Biostatistics Section, Department of Preventive and Societal Medicine, University of Nebraska Medical Center, Omaha, NE 68198. The work of both authors was supported in part by National Institute of Allergy and Infectious Diseases FIRST Award 1-R29-AI33466. The authors thank the developers of the diagnostics studied here for sharing their insights, experiences, and software, and Drs. Thomas Louis and Luke Tierney for helpful discussions and suggestions which greatly improved the manuscript. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #645046. 	[Title] Exponential Convergence of Langevin Diffusions and Their Discrete Approximations [Abstract] In this paper we consider a continous time method of approximating a given distribution using the Langevin diffusion dL t = dW t + 1 2 r log (L t )dt: We find conditions under which this diffusion converges exponentially quickly to or does not: in one dimension, these are essentially that for distributions with exponential tails of the form (x) / exp(fljxj fi ), 0 &lt; fi &lt; 1, exponential convergence occurs if and only if fi 1. We then consider conditions under which the discrete approximations to the diffusion converge. We first show that even when the diffusion itself converges, naive discretisations need not do so. We then consider a "Metropolis-adjusted" version of the algorithm, and find conditions under which this also converges at an exponential rate: perhaps surprisingly, even the Metropolised version need not converge exponentially fast even if the diffusion does. We briefly discuss a truncated form of the algorithm which, in practice, should avoid the difficulties of the other forms. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #646334. 	[Title] Self-Targeting Candidates for Metropolis-Hastings Algorithms [Abstract] The Metropolis-Hastings algorithm for estimating a distribution is based on choosing a candidate Markov chain and then accepting or rejecting moves of the candidate to produce a chain known to have as the invariant measure. The traditional methods use candidates essentially unconnected to . Based on diffusions for which is invariant, we develop for one-dimensional distributions a class of candidate distributions that "self-target" towards the high density areas of . These produce Metropolis-Hastings algorithms with convergence rates that appear to be considerably better than those known for the traditional candidate choices, such as random walk. In particular, for wide classes of these choices may effectively help reduce the "burn-in" problem. We illustrate this behaviour for examples with exponential and polynomial tails, and for a logistic regression model using a Gibbs sampling algorithm. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1131334. 	[Title] APPLICATIONS OF CHEEGER'S CONSTANT TO THE CONVERGENCE RATE OF MARKOV CHAINS ON R n [Abstract] Quantitative geometric rates of convergence for reversible Markov chains are closely related to the Cheeger's constant, which is hard to calculate for general state spaces. This article describes a geometric argument to bound the Cheeger's constant for chains on bounded subsets of R n . Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #642894. 	[Title] Rates of convergence of the Hastings and Metropolis algorithms [Abstract] We apply recent results in Markov chain theory to Hastings and Metropolis algorithms with either independent or symmetric candidate distributions, and provide necessary and sufficient conditions for the algorithms to converge at a geometric rate to a prescribed distribution . In the independence case (in IR k ) these indicate that geometric convergence essentially occurs if and only if the candidate density is bounded below by a multiple of ; in the symmetric case (in IR only) we show geometric convergence essentially occurs if and only if has geometric tails. We also evaluate recently developed computable bounds on the rates of convergence in this context: examples show that these theoretical bounds can be inherently extremely conservative, although when the chain is stochastically monotone the bounds may well be effective. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #126927. 	[Title] Convergence properties of perturbed Markov chains [Abstract] Acknowledgements. We thank Neal Madras, Radford Neal, Peter Rosenthal, and Richard Tweedie for helpful conversations. This work was partially supported by EPSRC of the U.K., and by NSERC of Canada. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #646357. 	[Title] Geometric and Subgeometric Convergence of Diffusions with Given Stationary Distributions, and Their Discretizations [Abstract] We describe algorithms for estimating a given measure known up to a constant of proportionality, based on a large class of diffusions (extending the Langevin model) for which is invariant. We show that under weak conditions one can choose from this class in such a way that the diffusions converge at exponential rate to , and one can even ensure that convergence is independent of the starting point of the algorithm. When convergence is less than exponential we show that it is often polynomial at known rates. We then consider methods of discretizing the diffusion in time, and find methods which inherit the convergence rates of the continuous time process. These contrast with the behaviour of the naive or Euler discretization, which can behave badly even in simple cases. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #644448. 	[Title] Adaptive Markov Chain Monte Carlo through Regeneration  Summary [Abstract] Markov chain Monte Carlo (MCMC) is used for evaluating expectations of functions of interest under a target distribution . This is done by calculating averages over the sample path of a Markov chain having as its stationary distribution. For computational efficiency, the Markov chain should be rapidly mixing. This can sometimes be achieved only by careful design of the transition kernel of the chain, on the basis of a detailed preliminary exploratory analysis of . An alternative approach might be to allow the transition kernel to adapt whenever new features of are encountered during the MCMC run. However, if such adaptation occurs infinitely often, the stationary distribution of the chain may be disturbed. We describe a framework, based on the concept of Markov chain regeneration, which allows adaptation to occur infinitely often, but which does not disturb the stationary distribution of the chain or the consistency of sample-path averages. Key Words: Adaptive method; Bayesian inference; Gibbs sampling; Markov chain Monte Carlo; Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #644577. 	[Title] Geometric Ergodicity and Hybrid Markov Chains [Abstract] Various notions of geometric ergodicity for Markov chains on general state spaces exist. In this paper, we review certain relations and implications among them. We then apply these results to a collection of chains commonly used in Markov chain Monte Carlo simulation algorithms, the so-called hybrid chains. We prove that under certain conditions, a hybrid chain will "inherit" the geometric ergodicity of its constituent parts. Acknowledgements. We thank Charlie Geyer for a number of very useful comments regarding spectral theory and central limit theorems. We thank Alison Gibbs, Phil Reiss, Peter Rosenthal, and Richard Tweedie for very helpful discussions. We thank the referee and the editor for many excellent suggestions. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1131236. 	[Title] Bounding Convergence Time of the Gibbs Sampler in Bayesian Image Restoration [Abstract] This paper gives precise, easy to compute bounds on the convergence time of the Gibbs sampler used in Bayesian image reconstruction. For sampling from the Gibbs distribution both with and without the presence of an external field, bounds that are N 2 in the number of pixels are obtained, with a proportionality constant that is easy to calculate. Some key words: Bayesian image restoration; Convergence; Gibbs sampler; Ising model; Markov chain Monte Carlo. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #642798. 	[Title] WEAK CONVERGENCE AND OPTIMAL SCALING OF RANDOM WALK METROPOLIS ALGORITHMS [Abstract] This paper considers the problem of scaling the proposal distribution of a multidimensional random walk Metropolis algorithm, in order to maximize the efficiency of the algorithm. The main result is a weak convergence result as the dimension of a sequence of target densities, n, converges to 1. When the proposal variance is appropriately scaled according to n, the sequence of stochastic processes formed by the first component of each Markov chain, converge to the appropriate limiting Langevin diffusion process. The limiting diffusion approximation admits a straight-forward efficiency maximization problem, and the resulting asymptotically optimal policy is related to the asymptotic acceptance rate of proposed moves for the algorithm. The asymptotically optimal acceptance rate is 0.234 under quite general conditions. The main result is proved in the case where the target density has a symmetric product form. Extensions of the result are discussed. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #645870. 	[Title] EXACT TRANSITION PROBABILITIES FOR THE INDEPENDENCE METROPOLIS SAMPLER [Abstract] A recent result of Jun Liu's has shown how to compute explicitly the eigen-values and eigenvectors for the Markov chain derived from a special case of the Hastings sampling algorithm, known as the indepdendence Metropolis sampler. In this note, we show how to extend the result to obtain exact n-step transition probabilities for any n. This is done first for a chain on a finite state space, and then extended to a general (discrete or continuous) state space. The paper concludes with some implications for diagnostic tests of convergence of Markov chain samplers. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1131257. 	[Title] Von Mises type statistics for single site updated local interaction random fields [Abstract] Random field models in image analysis and spatial statistics usually have local interactions. They can be simulated by Markov chains which update a single site at a time. The updating rules typically condition on only a few neighboring sites. If we want to approximate the expectation of a bounded function, can we make better use of the simulations than through the empirical estimator? We describe symmetrizations of the empirical estimator which are computationally feasible and can lead to considerable variance reduction. The method is reminiscent of the idea behind generalized von Mises statistics. To simplify the exposition, we consider mainly nearest neighbor random fields and the Gibbs sampler. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #644093. 	[Title] Possible biases induced by MCMC convergence diagnostics [Abstract]  Please summarise all the given information.

Paper #1131335 has the title `Estimating L 1 Error of Kernel Estimator: Monitoring Convergence of Markov Samplers  
` and the abstract `In many Markov chain Monte Carlo problems, the target density function is known up to a normalization constant. In this paper, we take advantage of this knowledge to facilitate the convergence diagnostic of a Markov sampler by estimating the L 1 error of a kernel estimator. Firstly, we propose an estimator of the normalization constant which is shown to be asymptotically normal under mixing and moment conditions. Secondly, the L 1 error of the kernel estimator is estimated using the normalization constant estimator, and the ratio of the estimated L 1 error to the true L 1 error is shown to converge to 1 in probability under similar conditions. Thirdly, we propose a sequential plot of the estimated L 1 error as a tool to monitor the convergence of the Markov sampler. Finally, a 2-dimensional bimodal example is given to illustrate the proposal, fl Bin Yu is Assistant Professor, Department of Statistics, University of California, Berkeley, CA 94720-3860. Research supported in part by the Junior Faculty Research Grant from University of California at Berkeley, grants DAAL03-91-G-007 and DAAH04-94-G-0232 from the Army Research Office, and grant DMS-9322817 from the National Science Foundation. The author is very grateful to Professors Peter Bickel and Andrew Gelman for many helpful discussions and their comments on the draft. Special thanks are due to Mr. Sam Buttrey for his help on simulation, to Professor Per Mykland and Mr. Karl Broman for commenting on the draft, and to two anonymous and two Markov samplers are compared in the example using the proposed diagnostic plot.
`.
Question: Which category should Paper #1131335 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #1131335 has the title `Estimating L 1 Error of Kernel Estimator: Monitoring Convergence of Markov Samplers  
` and the abstract `In many Markov chain Monte Carlo problems, the target density function is known up to a normalization constant. In this paper, we take advantage of this knowledge to facilitate the convergence diagnostic of a Markov sampler by estimating the L 1 error of a kernel estimator. Firstly, we propose an estimator of the normalization constant which is shown to be asymptotically normal under mixing and moment conditions. Secondly, the L 1 error of the kernel estimator is estimated using the normalization constant estimator, and the ratio of the estimated L 1 error to the true L 1 error is shown to converge to 1 in probability under similar conditions. Thirdly, we propose a sequential plot of the estimated L 1 error as a tool to monitor the convergence of the Markov sampler. Finally, a 2-dimensional bimodal example is given to illustrate the proposal, fl Bin Yu is Assistant Professor, Department of Statistics, University of California, Berkeley, CA 94720-3860. Research supported in part by the Junior Faculty Research Grant from University of California at Berkeley, grants DAAL03-91-G-007 and DAAH04-94-G-0232 from the Army Research Office, and grant DMS-9322817 from the National Science Foundation. The author is very grateful to Professors Peter Bickel and Andrew Gelman for many helpful discussions and their comments on the draft. Special thanks are due to Mr. Sam Buttrey for his help on simulation, to Professor Per Mykland and Mr. Karl Broman for commenting on the draft, and to two anonymous and two Markov samplers are compared in the example using the proposed diagnostic plot.
`.
Question: Which category should Paper #1131335 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].

This paper should be classified into: Probabilistic_Methods