Necessary conditions for geometric and polynomial ergodicity of random-walk-type

Jarner, Søren Fiig; Tweedie, Richard L.

doi:10.3150/bj/1066223269

Cited by 38 publications

(51 citation statements)

References 10 publications

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“…For later use we define the acceptance region A(x) ={y : (y) ≥ (x)} and the region of possible rejection R(x) ={y : (y) < (x)}. As shown in Jarner & Tweedie (2003) exponential or lighter tails of is necessary for geometric ergodicity of the random-walk Metropolis algorithm. In one dimension this is essentially also a sufficient condition (Mengersen & Tweedie, 1996) while in higher dimensions some extra case needs to be taken (Roberts & Tweedie, 1996b;Jarner & Hansen, 2000).…”

Section: Random-walk Metropolis Algorithmsmentioning

confidence: 99%

“…The polynomial rate of convergence in total variation of the algorithm is defined as the supremum of all for which (20) holds. We will use results of Jarner & Tweedie (2003) to show that the rates obtained are best possible. We first look at polynomial target densities on R + , then at non-symmetric polynomial target densities on R but with the same polynomial rate of decay in the two tails and finally at polynomial target densities on R d .…”

Section: Random-walk Metropolis Algorithmsmentioning

confidence: 99%

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Convergence of Heavy‐tailed Monte Carlo Markov Chain Algorithms

Jarner¹,

Roberts

2007

Scandinavian J Statistics

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In this paper, we use recent results of Jarner & Roberts ( "Ann. Appl. Probab.," 12, 2002, 224) to show polynomial convergence rates of Monte Carlo Markov Chain algorithms with polynomial target distributions, in particular random-walk Metropolis algorithms, Langevin algorithms and independence samplers. We also use similar methodology to consider polynomial convergence of the Gibbs sampler on a constrained state space. The main result for the random-walk Metropolis algorithm is that heavy-tailed proposal distributions lead to higher rates of convergence and thus to qualitatively better algorithms as measured, for instance, by the existence of central limit theorems for higher moments. Thus, the paper gives for the first time a theoretical justification for the common belief that heavy-tailed proposal distributions improve convergence in the context of random-walk Metropolis algorithms. Similar results are shown to hold for Langevin algorithms and the independence sampler, while results for the mixing of Gibbs samplers on uniform distributions on constrained spaces are rather different in character. Copyright 2007 Board of the Foundation of the Scandinavian Journal of Statistics..

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Section: Random-walk Metropolis Algorithmsmentioning

confidence: 99%

Section: Random-walk Metropolis Algorithmsmentioning

confidence: 99%

Convergence of Heavy‐tailed Monte Carlo Markov Chain Algorithms

Jarner¹,

Roberts

2007

Scandinavian J Statistics

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“…Conditions for geometric ergodicity in the case of generalised linear mixed models are given in [24], while spherically constrained target densities are discussed in [25]. In [26], the authors provide necessary conditions for the geometric convergence of RWM algorithms, which are related to the existence of exponential moments for π(·) and P (x, ·). Weaker forms of ergodicity and corresponding conditions are also discussed in the paper.…”

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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“…Markov chains with tight increments were analysed in [157], and sufficient conditions (in terms of the existence of moments) were given for polynomial ergodicity of a chain. This collaboration also led to two fundamental papers on the convergence of iterated random functions [145], [152], and related work on methods for simulation from a Dirichlet process in [156] based on [71].…”

Section: Applications To Queueing Theory and Time Seriesmentioning

confidence: 99%