Proximal MCMC for Bayesian Inference of Constrained and Regularized Estimation

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.