Almost sure convergence rates of adaptive increasingly rare Markov chain Monte Carlo
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917284466917376 |
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| author | Hofstadler, Julian Latuszynski, Krzysztof Roberts, Gareth O. Rudolf, Daniel |
| author_facet | Hofstadler, Julian Latuszynski, Krzysztof Roberts, Gareth O. Rudolf, Daniel |
| contents | We consider adaptive increasingly rare Markov chain Monte Carlo (MCMC) algorithms, which are adaptive MCMC methods, where the adaptation concerning the "past'' happens less and less frequently over time. Under a contraction assumption with respect to a Wasserstein-like function we deduce upper bounds of the convergence rate of Monte Carlo sums taking a renormalisation factor into account that is "almost'' the one that appears in a law of the iterated logarithm. We demonstrate the applicability of our results by considering different settings, among which are those of simultaneous geometric and uniform ergodicity. All proofs are carried out on an augmented state space, including the classical non-augmented setting as a special case. In contrast to other adaptive MCMC limit theory, some technical assumptions, like diminishing adaptation, are not needed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12122 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Almost sure convergence rates of adaptive increasingly rare Markov chain Monte Carlo Hofstadler, Julian Latuszynski, Krzysztof Roberts, Gareth O. Rudolf, Daniel Numerical Analysis Probability Statistics Theory 65C05, 65C20, 60J22 We consider adaptive increasingly rare Markov chain Monte Carlo (MCMC) algorithms, which are adaptive MCMC methods, where the adaptation concerning the "past'' happens less and less frequently over time. Under a contraction assumption with respect to a Wasserstein-like function we deduce upper bounds of the convergence rate of Monte Carlo sums taking a renormalisation factor into account that is "almost'' the one that appears in a law of the iterated logarithm. We demonstrate the applicability of our results by considering different settings, among which are those of simultaneous geometric and uniform ergodicity. All proofs are carried out on an augmented state space, including the classical non-augmented setting as a special case. In contrast to other adaptive MCMC limit theory, some technical assumptions, like diminishing adaptation, are not needed. |
| title | Almost sure convergence rates of adaptive increasingly rare Markov chain Monte Carlo |
| topic | Numerical Analysis Probability Statistics Theory 65C05, 65C20, 60J22 |
| url | https://arxiv.org/abs/2402.12122 |