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Perfect sampling from independent Metropolis-hastings chains

JOURNAL OF STATISTICAL PLANNING AND INFERENCE 2002年第2期104卷 297-314页

作者： Corcoran, JN Tweedie, RL Univ Colorado Dept Appl Math Boulder CO 80309 USA

"Perfect sampling" enables exact draws from the invariant measure of a Markov chain. We show that the independent Metropolis-hastings chain has certain stochastic monotonicity properties that enable a perfect sampling algorithm to be implemented, at least when the candidate is overdispersed with respect to the target distribution. We prove that the algorithm has an optimal geometric convergence rate, and applications show that it may converge extreme rapidly. (C) 2002 Elsevier Science B.V. All rights reserved.

关键词： metropolis algorithms hastings algorithms Markov chains invariant measures backwards coupling coupling from the past exact sampling perfect sampling

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A note on acceptance rate criteria for CLTs for Metropolis-hastings algorithms

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JOURNAL OF APPLIED PROBABILITY 1999年第4期36卷 1210-1217页

作者： Roberts, GO Univ Cambridge Cambridge CB2 1TN England

This paper 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 specialized, although, in both cases, the problems which arise are typical of problems commonly occurring for the particular algorithm being used.

关键词： hastings algorithms Metropolis algorithms Markov chain Monte Carlo central limit theorems

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Rates of convergence of the hastings and Metropolis algorithms

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ANNALS OF STATISTICS 1996年第1期24卷 101-121页

作者： Mengersen, KL Tweedie, RL COLORADO STATE UNIV DEPT STATFT COLLINSCO 80523

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 pi. In the independence case (in R(k)) these indicate that geometric convergence essentially occurs if and only if the candidate density is bounded below by a multiple of pi;in the symmetric case (in R only) we show geometric convergence essentially occurs if and only if pi 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.

关键词： posterior distributions hastings algorithms Metropolis algorithms Gibbs sampling Markov chain Monte Carlo irreducible Markov processes geometric ergodicity stochastic monotonicity log-concave distributions

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A Coupling-Regeneration Scheme for Diagnosing Convergence in Markov Chain Monte Carlo algorithms

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Quarterly Publications of the American Statistical Association 1998年第441期93卷 238-248页

Here I propose a convergence diagnostic for Markov chain Monte Carlo (MCMC) algorithms based on couplings of a Markov chain with an auxiliary chain that is periodically restarted from a fixed parameter value. The diagnostic provides a mechanism for estimating the specific constants governing the rate of convergence of geometrically and uniformly ergodic chains, and provides a lower bound on the effective sample size of a MCMC run. It also provides a simple procedure for obtaining what is, with high probability, an independent sample from the stationary distribution.

关键词： Gibbs sampling hastings algorithms Maximal coupling Metropolis algorithm Regeneration and coupling

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Exponential convergence of Langevin distributions and their discrete approximations

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Bernoulli 1996年第4期2.0卷 341-363页

作者： Roberts, Gareth O. Tweedie, Richard L. Statistical Laboratory Department of Pure Mathematics and Mathematical Statistics 16 Mill Lane Cambridge CB2 1SB United Kingdom Department of Statistics Colorado State University Fort Collins 80523 CO United States

In this paper we consider a continuous-time method of approximating a given distribution π using the Langevin diffusion dLt=dWt+1/2∇log π(Lt)dt. We find conditions under this diffusion converges exponentially quickl... 详细信息

In this paper we consider a continuous-time method of approximating a given distribution π using the Langevin diffusion dL_t=dW_t+1/2∇log π(L_t)dt. We find conditions under this diffusion converges exponentially quickly to π or does not: In one dimension, these are essentially that for distributions with exponential tails of the form π(x) ∝ exp (-γ|x|^β) 0<β<∞, exponential onvergence occurs if and only if β≥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 discretizations 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 Metropolized 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. © 1996 Chapman & Hall.

关键词： Diffusions Discrete approximations Geometric ergodicity hastings algorithms Irreducible Markov processes Langevin models Markov chain Monte Carlo Metropolis algorithms Posterior distributions

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BAYESIAN COMPUTATION AND STOCHASTIC-SYSTEMS

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STATISTICAL SCIENCE 1995年第1期10卷 3-41页

作者： BESAG, J GREEN, P HIGDON, D MENGERSEN, K UNIV WASHINGTON CTR SPATIAL STATSEATTLEWA 98195 UNIV BRISTOL DEPT MATHBRISTOL BS8 1TWAVONENGLAND QUEENSLAND UNIV TECHNOL SCH MATHBRISBANEQLD 4001AUSTRALIA

Markov chain Monte Carlo (MCMC) methods have been used extensively in statistical physics over the last 40 years, in spatial statistics for the past 20 and in Bayesian image analysis over the last decade. In the last five years, MCMC has been introduced into significance testing, general Bayesian inference and maximum likelihood estimation. This paper presents basic methodology of MCMC, emphasizing the Bayesian paradigm, conditional probability and the intimate relationship with Markov random fields in spatial statistics. hastings algorithms are discussed, including Gibbs, Metropolis and some other variations. Pairwise difference priors are described and are used subsequently in three Bayesian applications, in each of which there is a pronounced spatial or temporal aspect to the modeling. The examples involve logistic regression in the presence of unobserved covariates and ordinal factors;the analysis of agricultural field experiments, with adjustment for fertility gradients;and processing of low-resolution medical images obtained by a gamma camera. Additional methodological issues arise in each of these applications and in the Appendices. The paper lays particular emphasis on the calculation of posterior probabilities and concurs with others in its view that MCMC facilitates a fundamental breakthrough in applied Bayesian modeling.

关键词： AGRICULTURAL FIELD EXPERIMENTS BAYESIAN INFERENCE CONDITIONAL DISTRIBUTIONS DECONVOLUTION GAMMA-CAMERA IMAGING GIBBS SAMPLER hastings algorithms IMAGE ANALYSIS LOGISTIC REGRESSION MARKOV CHAIN MONTE CARLO MARKOV RANDOM FIELDS METROPOLIS METHOD PROSTATE CANCER SIMULTANEOUS CREDIBLE REGIONS SPATIAL STATISTICS TIME REVERSIBILITY UNOBSERVED COVARIATES VARIETY TRIALS

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