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Perturbation bounds for Monte Carlo within Metropolis via restricted approximations
Felipe Medina-Aguayo1, Daniel Rudolf2, Nikolaus Schweizer3
1Department of Mathematics and Statistics, University of Reading Whiteknights, PO Box 220, Reading RG6 6AX, United Kingdom.
Stochastic Processes and Their Applications
|April 8, 2020
Summary
The Monte Carlo within Metropolis (MCwM) algorithm offers approximate sampling for complex distributions. This study provides bounds on its accuracy compared to the Metropolis-Hastings (MH) algorithm, crucial for computational statistics.
Area of Science:
- Computational Statistics
- Markov Chain Monte Carlo Methods
Background:
- Intractable target distributions pose challenges in statistical inference.
- The Metropolis-Hastings (MH) algorithm is a standard Markov chain Monte Carlo (MCMC) method.
- Approximate sampling methods are needed when exact MCMC is infeasible.
Purpose of the Study:
- To analyze the Monte Carlo within Metropolis (MCwM) algorithm as a perturbed MH algorithm.
- To derive explicit bounds on the difference between MCwM and MH chain distributions.
- To establish theoretical underpinnings for approximate MCMC methods.
Main Methods:
- Interpreting MCwM as a perturbed Metropolis-Hastings (MH) algorithm.
- Utilizing geometric ergodicity assumptions for the unperturbed Markov chain.
- Developing novel perturbation results for Markov chains.
- Deriving explicit estimates for the difference between perturbed and unperturbed chain distributions.
Main Results:
- Explicit bounds were derived for the difference between the k-th step distributions of MCwM and MH chains.
- The bounds depend on controlling transition probability differences and verifying perturbed chain stability.
- Novel perturbation results for Markov chains were established.
Conclusions:
- The derived bounds provide theoretical guarantees for the accuracy of the MCwM algorithm.
- The perturbation results have broader applicability in Markov chain analysis.
- Controlling transition probability differences and ensuring stability are key for applying these bounds.
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