تقرير
Parallel MCMC Algorithms: Theoretical Foundations, Algorithm Design, Case Studies
| العنوان: | Parallel MCMC Algorithms: Theoretical Foundations, Algorithm Design, Case Studies |
|---|---|
| المؤلفون: | Glatt-Holtz, Nathan E., Holbrook, Andrew J., Krometis, Justin A., Mondaini, Cecilia F. |
| سنة النشر: | 2022 |
| المجموعة: | Mathematics Statistics |
| مصطلحات موضوعية: | Statistics - Computation, Mathematics - Probability, Mathematics - Statistics Theory, 62D05, 60J22, 65C05, 65Y05 |
| الوصف: | Parallel Markov Chain Monte Carlo (pMCMC) algorithms generate clouds of proposals at each step to efficiently resolve a target probability distribution. We build a rigorous foundational framework for pMCMC algorithms that situates these methods within a unified 'extended phase space' measure-theoretic formalism. Drawing on our recent work that provides a comprehensive theory for reversible single proposal methods, we herein derive general criteria for multiproposal acceptance mechanisms which yield ergodic chains on general state spaces. Our formulation encompasses a variety of methodologies, including proposal cloud resampling and Hamiltonian methods, while providing a basis for the derivation of novel algorithms. In particular, we obtain a top-down picture for a class of methods arising from 'conditionally independent' proposal structures. As an immediate application, we identify several new algorithms including a multiproposal version of the popular preconditioned Crank-Nicolson (pCN) sampler suitable for high- and infinite-dimensional target measures which are absolutely continuous with respect to a Gaussian base measure. To supplement our theoretical results, we carry out a selection of numerical case studies that evaluate the efficacy of these novel algorithms. First, noting that the true potential of pMCMC algorithms arises from their natural parallelizability, we provide a limited parallelization study using TensorFlow and a graphics processing unit to scale pMCMC algorithms that leverage as many as 100k proposals at each step. Second, we use our multiproposal pCN algorithm (mpCN) to resolve a selection of problems in Bayesian statistical inversion for partial differential equations motivated by fluid measurement. These examples provide preliminary evidence of the efficacy of mpCN for high-dimensional target distributions featuring complex geometries and multimodal structures. Comment: Minor revisions from previous version |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2209.04750 |
| رقم الأكسشن: | edsarx.2209.04750 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |