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| Format: | Preprint |
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2019
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| Online Access: | https://arxiv.org/abs/1907.11973 |
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| _version_ | 1866914086131859456 |
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| author | Birrell, Jeremiah Rey-Bellet, Luc |
| author_facet | Birrell, Jeremiah Rey-Bellet, Luc |
| contents | In this work we provide performance guarantees for hypocoercive non-reversible MCMC samplers $X_t$ with invariant measure $μ_*$; our results apply in particular to the Langevin equation, Hamiltonian Monte-Carlo, and the bouncy particle and zig-zag samplers. Specifically, we establish a concentration inequality of Bernstein type for ergodic averages $\frac{1}{T} \int_0^T f(X_t)\, dt$. As a consequence we provide two types of performance guarantees: (a) explicit non-asymptotic confidence intervals for $\int f dμ_*$ when using a finite time ergodic average with given initial condition $μ$ and (b) uncertainty quantification (UQ) bounds, expressed in terms of relative entropy rate, on the bias of $\int f dμ_*$ when using an alternative or approximate processes $\widetilde{X}_t$. (Results in (b) generalize results (arXiv:1812.05174) from the authors for coercive dynamics.) The concentration inequality is proved by combining the approach via Feynman-Kac semigroups first noted by Wu with the hypocoercive estimates of Dolbeault, Mouhot and Schmeiser (arXiv:1005.1495) developed for the Langevin equation and generalized to partially deterministic Markov processes by Andrieu et al. (arXiv:1808.08592). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1907_11973 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Concentration Inequalities and UQ Bounds for Hypocoercive MCMC Samplers Birrell, Jeremiah Rey-Bellet, Luc Probability In this work we provide performance guarantees for hypocoercive non-reversible MCMC samplers $X_t$ with invariant measure $μ_*$; our results apply in particular to the Langevin equation, Hamiltonian Monte-Carlo, and the bouncy particle and zig-zag samplers. Specifically, we establish a concentration inequality of Bernstein type for ergodic averages $\frac{1}{T} \int_0^T f(X_t)\, dt$. As a consequence we provide two types of performance guarantees: (a) explicit non-asymptotic confidence intervals for $\int f dμ_*$ when using a finite time ergodic average with given initial condition $μ$ and (b) uncertainty quantification (UQ) bounds, expressed in terms of relative entropy rate, on the bias of $\int f dμ_*$ when using an alternative or approximate processes $\widetilde{X}_t$. (Results in (b) generalize results (arXiv:1812.05174) from the authors for coercive dynamics.) The concentration inequality is proved by combining the approach via Feynman-Kac semigroups first noted by Wu with the hypocoercive estimates of Dolbeault, Mouhot and Schmeiser (arXiv:1005.1495) developed for the Langevin equation and generalized to partially deterministic Markov processes by Andrieu et al. (arXiv:1808.08592). |
| title | Concentration Inequalities and UQ Bounds for Hypocoercive MCMC Samplers |
| topic | Probability |
| url | https://arxiv.org/abs/1907.11973 |