scholarly journals Ergodicity of Markov chain Monte Carlo with reversible proposal

2017 ◽  
Vol 54 (2) ◽  
pp. 638-654 ◽  
Author(s):  
K. Kamatani
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Random Walk ◽  
Markov Chain Monte Carlo ◽  
Geometric Ergodicity ◽  
Suitable Transformation ◽  
Ergodic Properties ◽  
Random Walk Metropolis ◽  
Heavy Tailed ◽  
Random Walk Metropolis Algorithm

Abstract We describe the ergodic properties of some Metropolis–Hastings algorithms for heavy-tailed target distributions. The results of these algorithms are usually analyzed under a subgeometric ergodic framework, but we prove that the mixed preconditioned Crank–Nicolson (MpCN) algorithm has geometric ergodicity even for heavy-tailed target distributions. This useful property comes from the fact that, under a suitable transformation, the MpCN algorithm becomes a random-walk Metropolis algorithm.

scholarly journals Markov Chain Monte Carlo for Computing Rare-Event Probabilities for a Heavy-Tailed Random Walk

2014 ◽  
Vol 51 (2) ◽  
pp. 359-376 ◽  
Author(s):  
Thorbjörn Gudmundsson ◽  
Henrik Hult
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Random Walk ◽  
Markov Chain Monte Carlo ◽  
Conditional Distribution ◽  
Rare Event ◽  
High Threshold ◽  
Full Generality ◽  
Normalizing Constant ◽  
Heavy Tailed

In this paper a method based on a Markov chain Monte Carlo (MCMC) algorithm is proposed to compute the probability of a rare event. The conditional distribution of the underlying process given that the rare event occurs has the probability of the rare event as its normalizing constant. Using the MCMC methodology, a Markov chain is simulated, with the aforementioned conditional distribution as its invariant distribution, and information about the normalizing constant is extracted from its trajectory. The algorithm is described in full generality and applied to the problem of computing the probability that a heavy-tailed random walk exceeds a high threshold. An unbiased estimator of the reciprocal probability is constructed whose normalized variance vanishes asymptotically. The algorithm is extended to random sums and its performance is illustrated numerically and compared to existing importance sampling algorithms.

scholarly journals Markov Chain Monte Carlo for Computing Rare-Event Probabilities for a Heavy-Tailed Random Walk

2014 ◽  
Vol 51 (02) ◽  
pp. 359-376 ◽  
Author(s):  
Thorbjörn Gudmundsson ◽  
Henrik Hult
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Random Walk ◽  
Markov Chain Monte Carlo ◽  
Conditional Distribution ◽  
Rare Event ◽  
High Threshold ◽  
Full Generality ◽  
Normalizing Constant ◽  
Heavy Tailed

In this paper a method based on a Markov chain Monte Carlo (MCMC) algorithm is proposed to compute the probability of a rare event. The conditional distribution of the underlying process given that the rare event occurs has the probability of the rare event as its normalizing constant. Using the MCMC methodology, a Markov chain is simulated, with the aforementioned conditional distribution as its invariant distribution, and information about the normalizing constant is extracted from its trajectory. The algorithm is described in full generality and applied to the problem of computing the probability that a heavy-tailed random walk exceeds a high threshold. An unbiased estimator of the reciprocal probability is constructed whose normalized variance vanishes asymptotically. The algorithm is extended to random sums and its performance is illustrated numerically and compared to existing importance sampling algorithms.

On the geometric ergodicity of hybrid samplers

2003 ◽  
Vol 40 (1) ◽  
pp. 123-146 ◽  
Author(s):  
G. Fort ◽  
E. Moulines ◽  
G. O. Roberts ◽  
J. S. Rosenthal
Keyword(s):  
Random Walk ◽  
Sufficient Conditions ◽  
Metropolis Algorithm ◽  
Geometric Ergodicity ◽  
Target Density ◽  
Symmetric Random Walk ◽  
Uniform Ergodicity ◽  
Random Walk Metropolis ◽  
Random Walk Metropolis Algorithm

In this paper, we consider the random-scan symmetric random walk Metropolis algorithm (RSM) on ℝd. This algorithm performs a Metropolis step on just one coordinate at a time (as opposed to the full-dimensional symmetric random walk Metropolis algorithm, which proposes a transition on all coordinates at once). We present various sufficient conditions implying V-uniform ergodicity of the RSM when the target density decreases either subexponentially or exponentially in the tails.

On the geometric ergodicity of hybrid samplers

2003 ◽  
Vol 40 (01) ◽  
pp. 123-146 ◽  
Author(s):  
G. Fort ◽  
E. Moulines ◽  
G. O. Roberts ◽  
J. S. Rosenthal
Keyword(s):  
Random Walk ◽  
Sufficient Conditions ◽  
Metropolis Algorithm ◽  
Geometric Ergodicity ◽  
Target Density ◽  
Symmetric Random Walk ◽  
Uniform Ergodicity ◽  
Random Walk Metropolis ◽  
Random Walk Metropolis Algorithm

In this paper, we consider the random-scan symmetric random walk Metropolis algorithm (RSM) on ℝ d . This algorithm performs a Metropolis step on just one coordinate at a time (as opposed to the full-dimensional symmetric random walk Metropolis algorithm, which proposes a transition on all coordinates at once). We present various sufficient conditions implying V-uniform ergodicity of the RSM when the target density decreases either subexponentially or exponentially in the tails.

scholarly journals Component-Wise Markov Chain Monte Carlo: Uniform and Geometric Ergodicity under Mixing and Composition

2013 ◽  
Vol 28 (3) ◽  
pp. 360-375 ◽  
Author(s):  
Alicia A. Johnson ◽  
Galin L. Jones ◽  
Ronald C. Neath
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Geometric Ergodicity

scholarly journals IntCal09 and Marine09 Radiocarbon Age Calibration Curves, 0–50,000 Years cal BP

Radiocarbon ◽  
2009 ◽  
Vol 51 (4) ◽  
pp. 1111-1150 ◽  
Author(s):  
P J Reimer ◽  
M G L Baillie ◽  
E Bard ◽  
A Bayliss ◽  
J W Beck ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Random Walk ◽  
Markov Chain Monte Carlo ◽  
Working Group ◽  
Random Walk Model ◽  
Data Sets ◽  
Radiocarbon Age ◽  
Calibration Curves ◽  
Radiocarbon Calibration

The IntCal04 and Marine04 radiocarbon calibration curves have been updated from 12 cal kBP (cal kBP is here defined as thousands of calibrated years before AD 1950), and extended to 50 cal kBP, utilizing newly available data sets that meet the IntCal Working Group criteria for pristine corals and other carbonates and for quantification of uncertainty in both the 14C and calendar timescales as established in 2002. No change was made to the curves from 0–12 cal kBP. The curves were constructed using a Markov chain Monte Carlo (MCMC) implementation of the random walk model used for IntCal04 and Marine04. The new curves were ratified at the 20th International Radiocarbon Conference in June 2009 and are available in the Supplemental Material at www.radiocarbon.org.

A Gaussian Mixture Model as a Proposal Distribution for Efficient Markov-Chain Monte Carlo Characterization of Uncertainty in Reservoir Description and Forecasting

SPE Journal ◽  
2019 ◽  
Vol 25 (01) ◽  
pp. 001-036 ◽  
Author(s):  
Xin Li ◽  
Albert C. Reynolds
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Random Walk ◽  
Markov Chain Monte Carlo ◽  
Uncertainty Quantification ◽  
Gaussian Mixture Model ◽  
Gaussian Mixture ◽  
Mcmc Algorithm ◽  
Proposal Distribution

Summary Generating an estimate of uncertainty in production forecasts has become nearly standard in the oil industry, but is often performed with procedures that yield at best a highly approximate uncertainty quantification. Formally, the uncertainty quantification of a production forecast can be achieved by generating a correct characterization of the posterior probability-density function (PDF) of reservoir-model parameters conditional to dynamic data and then sampling this PDF correctly. Although Markov-chain Monte Carlo (MCMC) provides a theoretically rigorous method for sampling any target PDF that is known up to a normalizing constant, in reservoir-engineering applications, researchers have found that it might require extraordinarily long chains containing millions to hundreds of millions of states to obtain a correct characterization of the target PDF. When the target PDF has a single mode or has multiple modes concentrated in a small region, it might be possible to implement a proposal distribution dependent on a random walk so that the resulting MCMC algorithm derived from the Metropolis-Hastings acceptance probability can yield a good characterization of the posterior PDF with a computationally feasible chain length. However, for a high-dimensional multimodal PDF with modes separated by large regions of low or zero probability, characterizing the PDF with MCMC using a random walk is not computationally feasible. Although methods such as population MCMC exist for characterizing a multimodal PDF, their computational cost generally makes the application of these algorithms far too costly for field application. In this paper, we design a new proposal distribution using a Gaussian mixture PDF for use in MCMC where the posterior PDF can be multimodal with the modes spread far apart. Simply put, the method generates modes using a gradient-based optimization method and constructs a Gaussian mixture model (GMM) to use as the basic proposal distribution. Tests on three simple problems are presented to establish the validity of the method. The performance of the new MCMC algorithm is compared with that of random-walk MCMC and is also compared with that of population MCMC for a target PDF that is multimodal.

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