Geometric ergodicity of Metropolis algorithms

Jarner, Søren Fiig; Hansen, E. T.

doi:10.1016/s0304-4149(99)00082-4

Cited by 161 publications

(182 citation statements)

References 17 publications

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“…These techniques are random variable generators that allow us to draw samples from complicated distributions. Perhaps the most commonly used MCMC algorithm is the Metropolis-Hastings (MH), which operates as follows [14]: from a given proposal distribution, we construct an irreducible Markov chain whose stationary distribution is the sought posterior law (i.e., samples generated by the algorithm after a suitable burn-in period are distributed according to desired posterior law). At each iteration t, a decision rule is applied to accept or reject the proposed sample given by the following acceptance probability:…”

Section: Introductionmentioning

confidence: 99%

An Auxiliary Variable Method for Markov Chain Monte Carlo Algorithms in High Dimension

Marnissi

Chouzenoux

Benazza-Benyahia

et al. 2018

Entropy

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Abstract:In this paper, we are interested in Bayesian inverse problems where either the data fidelity term or the prior distribution is Gaussian or driven from a hierarchical Gaussian model. Generally, Markov chain Monte Carlo (MCMC) algorithms allow us to generate sets of samples that are employed to infer some relevant parameters of the underlying distributions. However, when the parameter space is high-dimensional, the performance of stochastic sampling algorithms is very sensitive to existing dependencies between parameters. In particular, this problem arises when one aims to sample from a high-dimensional Gaussian distribution whose covariance matrix does not present a simple structure. Another challenge is the design of Metropolis-Hastings proposals that make use of information about the local geometry of the target density in order to speed up the convergence and improve mixing properties in the parameter space, while not being too computationally expensive. These two contexts are mainly related to the presence of two heterogeneous sources of dependencies stemming either from the prior or the likelihood in the sense that the related covariance matrices cannot be diagonalized in the same basis. In this work, we address these two issues. Our contribution consists of adding auxiliary variables to the model in order to dissociate the two sources of dependencies. In the new augmented space, only one source of correlation remains directly related to the target parameters, the other sources of correlations being captured by the auxiliary variables. Experiments are conducted on two practical image restoration problems-namely the recovery of multichannel blurred images embedded in Gaussian noise and the recovery of signal corrupted by a mixed Gaussian noise. Experimental results indicate that adding the proposed auxiliary variables makes the sampling problem simpler since the new conditional distribution no longer contains highly heterogeneous correlations. Thus, the computational cost of each iteration of the Gibbs sampler is significantly reduced while ensuring good mixing properties.

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Section: Introductionmentioning

confidence: 99%

An Auxiliary Variable Method for Markov Chain Monte Carlo Algorithms in High Dimension

Marnissi

Chouzenoux

Benazza-Benyahia

et al. 2018

Entropy

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“…The following lemma forms the basis for proving the validity of extended Gibbs sampling as an MCMC sampler. (For a discussion of ergodicity of MCMC samplers, see Roberts and Rosenthal 1999;Jarner and Hansen 2000. ) …”

Section: Extended Gibbs Samplingmentioning

confidence: 99%

Multivariate-From-Univariate MCMC Sampler: The R Package MfUSampler

Mahani¹,

Sharabiani²

2017

J. Stat. Soft.

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The R package MfUSampler provides Markov chain Monte Carlo machinery for generating samples from multivariate probability distributions using univariate sampling algorithms such as the slice sampler and the adaptive rejection sampler. The multivariate wrapper performs a full cycle of univariate sampling steps, one coordinate at a time. In each step, the latest sample values obtained for other coordinates are used to form the conditional distributions. The concept is an extension of Gibbs sampling where each step involves, not an independent sample from the conditional distribution, but a Markov transition for which the conditional distribution is invariant. The software relies on proportionality of conditional distributions to the joint distribution to implement a thin wrapper for producing conditionals. Examples illustrate basic usage as well as methods for improving performance. By encapsulating the multivariate-from-univariate logic, package MfUSampler provides a reliable package for rapid prototyping of custom Bayesian models while allowing for incremental performance optimizations such as taking advantage of conditional independence, and high-performance implementation of function evaluations. Utility functions for MCMC diagnostics as well as sample-based construction of predictive posterior distributions are provided in MfUSampler.

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“…Ergodicity results for a Markov chain constructed using the RWM algorithm also exist [21][22][23]. At least exponentially light tails are a necessity for π(x) for geometric ergodicity, which means that π(x)/e − x → c as x → ∞, for some constant, c. For super-exponential tails (where π(x) → 0 at a faster than the exponential rate), additional conditions are required [21,23].…”

Section: Random Walk Proposalsmentioning

confidence: 99%

“…At least exponentially light tails are a necessity for π(x) for geometric ergodicity, which means that π(x)/e − x → c as x → ∞, for some constant, c. For super-exponential tails (where π(x) → 0 at a faster than the exponential rate), additional conditions are required [21,23]. We demonstrate with a simple example why heavy-tailed forms of π(x) pose difficulties here (where π(x) → 0 at a rate slower than e − x ).…”

Section: Random Walk Proposalsmentioning

confidence: 99%

Information-Geometric Markov Chain Monte Carlo Methods Using Diffusions

Livingstone

Girolami

2014

Entropy

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Recent work incorporating geometric ideas in Markov chain Monte Carlo is reviewed in order to highlight these advances and their possible application in a range of domains beyond statistics. A full exposition of Markov chains and their use in Monte Carlo simulation for statistical inference and molecular dynamics is provided, with particular emphasis on methods based on Langevin diffusions. After this, geometric concepts in Markov chain Monte Carlo are introduced. A full derivation of the Langevin diffusion on a Riemannian manifold is given, together with a discussion of the appropriate Riemannian metric choice for different problems. A survey of applications is provided, and some open questions are discussed.

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Geometric ergodicity of Metropolis algorithms

Cited by 161 publications

References 17 publications

An Auxiliary Variable Method for Markov Chain Monte Carlo Algorithms in High Dimension

An Auxiliary Variable Method for Markov Chain Monte Carlo Algorithms in High Dimension

Multivariate-From-Univariate MCMC Sampler: The R Package MfUSampler

Information-Geometric Markov Chain Monte Carlo Methods Using Diffusions

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