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Abstract When implementing Markov Chain Monte Carlo (MCMC) algorithms, perturbation caused by numerical errors is sometimes inevitable. This paper studies how the perturbation of MCMC affects the convergence speed and approximation accuracy. Our results show that when the original Markov chain converges to stationarity fast enough and the perturbed transition kernel is a good approximation to the original transition kernel, the corresponding perturbed sampler has fast convergence speed and high approximation accuracy as well. Our convergence analysis is conducted under either the Wasserstein metric or the$$\chi^2$$metric, both are widely used in the literature. The results can be extended to obtain non-asymptotic error bounds for MCMC estimators. We demonstrate how to apply our convergence and approximation results to the analysis of specific sampling algorithms, including Random walk Metropolis, Metropolis adjusted Langevin algorithm with perturbed target densities, and parallel tempering Monte Carlo with perturbed densities. Finally, we present some simple numerical examples to verify our theoretical claims.
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Malliavin-Based Multilevel Monte Carlo Estimators for Densities of Max-Stable Processes
We introduce a class of unbiased Monte Carlo estimators for multivariate densities of max-stable fields generated by Gaussian processes. Our estimators take advantage of recent results on the exact simulation of max-stable fields combined with identities studied in the Malliavin calculus literature and ideas developed in the multilevel Monte Carlo literature. Our approach allows estimating multivariate densities of max-stable fields with precision eps at a computational cost of order O(eps{−2}*logloglog(1/eps)).
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- Award ID(s):
- 1838576
- PAR ID:
- 10074255
- Date Published:
- Journal Name:
- Monte Carlo and Quasi-Monte Carlo Methods 2016
- Page Range / eLocation ID:
- 75-97
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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