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Proximal Markov Chain Monte Carlo (ProxMCMC) offers a flexible Bayesian inference framework for complex estimation problems. This enhanced method allows data-adaptive parameter estimation and scales to high-dimensional data using advanced sampling algorithms.

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Hamiltonian Monte CarloMoreau-Yosida envelopeProximal mapping

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Area of Science:

  • Computational Statistics
  • Bayesian Inference
  • Machine Learning

Background:

  • Proximal Markov Chain Monte Carlo (ProxMCMC) was initially developed for Bayesian imaging.
  • Existing ProxMCMC methods used fixed parameters and the Langevin algorithm.
  • Constrained and regularized estimation pose challenges in both frequentist and Bayesian statistics.

Purpose of the Study:

  • To extend ProxMCMC into a fully Bayesian framework.
  • To enable data-adaptive estimation of all parameters, including regularization strength.
  • To enhance scalability for high-dimensional problems.

Main Methods:

  • Developed a fully Bayesian ProxMCMC by incorporating data-adaptive parameter estimation.
  • Utilized Moreau-Yosida envelope for smooth approximation of total-variation regularization.
  • Employed advanced sampling algorithms like Hamiltonian Monte Carlo for improved scalability.

Main Results:

  • Demonstrated the versatility of ProxMCMC across various statistical estimation tasks.
  • Showcased the framework's ability to handle problems previously considered intractable.
  • Validated the effectiveness of data-adaptive parameter estimation in ProxMCMC.

Conclusions:

  • ProxMCMC provides a powerful and modular Bayesian inference approach.
  • The extended framework addresses limitations of previous ProxMCMC implementations.
  • ProxMCMC is applicable to a wide range of challenging statistical and machine learning problems.