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Parallel MCMC algorithms: theoretical foundations, algorithm design, case studies.

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Summary
This summary is machine-generated.

This study introduces a unified framework for parallel Markov Chain Monte Carlo (pMCMC) algorithms, developing new multiproposal methods like the multiproposal preconditioned Crank-Nicolson (mpCN) sampler for complex Bayesian inference problems.

Keywords:
Bayesian statistical inversionHamiltonian Monte Carlo (HMC)Markov Chain Monte Carlo (pMCMC)Metropolis–Hastings kernelsParallel (Multiproposal)graphics processing units (GPU)high-performance computingpreconditioned Crank–Nicolson (pCN)simplicial sampler

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Area of Science:

  • Computational Statistics
  • Bayesian Inference
  • Markov Chain Monte Carlo Methods

Background:

  • Parallel Markov Chain Monte Carlo (pMCMC) algorithms are crucial for efficiently exploring complex probability distributions.
  • Existing theories for single-proposal methods lack a unified framework for multiproposal extensions.
  • The need for scalable and efficient algorithms in high-dimensional Bayesian inference is growing.

Purpose of the Study:

  • To establish a rigorous, measure-theoretic framework for parallel Markov Chain Monte Carlo (pMCMC) algorithms.
  • To derive general criteria for multiproposal acceptance mechanisms ensuring ergodicity.
  • To develop novel pMCMC algorithms, including a multiproposal preconditioned Crank-Nicolson (mpCN) sampler.

Main Methods:

  • Development of an 'extended phase space' formalism for pMCMC algorithms.
  • Derivation of general criteria for multiproposal acceptance mechanisms.
  • Identification and application of new algorithms, including a multiproposal preconditioned Crank-Nicolson (mpCN) sampler.
  • Numerical case studies involving parallelization and Bayesian statistical inversion.

Main Results:

  • A unified theoretical framework for pMCMC algorithms, encompassing various existing and novel methods.
  • General criteria for multiproposal acceptance mechanisms that guarantee ergodic chains.
  • Introduction of novel algorithms, notably the multiproposal preconditioned Crank-Nicolson (mpCN) sampler.
  • Demonstration of mpCN's efficacy in high-dimensional Bayesian inversion problems with complex target distributions.

Conclusions:

  • The proposed framework provides a solid theoretical foundation for pMCMC methods.
  • The novel mpCN algorithm shows significant promise for tackling challenging high-dimensional Bayesian inference tasks.
  • pMCMC algorithms, particularly mpCN, are well-suited for parallel computing architectures and modern high-performance computing.