Bayesian Inference Using the Proximal Mapping: Uncertainty Quantification Under Varying Dimensionality.

Area of Science:

• Statistics
• Bayesian Inference
• Machine Learning

Background:

• Statistical applications often involve parameters in varying or unknown dimensional spaces, posing challenges for uncertainty quantification.
• Traditional Bayesian methods struggle with assigning priors for unknown dimensions, often requiring complex, combinatorial dimension-selection priors.
• Frequentist regularization techniques like fused lasso and nuclear norm penalty are effective for point estimation but lack probabilistic uncertainty estimation.

Purpose of the Study:

• To develop a new Bayesian generative process for priors that accommodates varying or unknown dimensional spaces.
• To enable principled probabilistic uncertainty estimation for models with unknown dimensions.
• To integrate popular frequentist regularization methods and algorithms within a Bayesian framework.

Main Methods:

• Proposed a novel generative process for priors starting from continuous random variables (e.g., multivariate Gaussian).
• Utilized proximal mapping to transform variables into a varying-dimensional space, creating a new class of Bayesian models.
• Leveraged geometric measure theory for theoretical justification and Hamiltonian Monte Carlo for posterior computation.

Main Results:

• Developed a flexible Bayesian framework that directly incorporates frequentist regularization techniques (e.g., nuclear norm penalty).
• Demonstrated that the proposed method provides principled and probabilistic uncertainty estimation.
• Showcased the framework's applicability through an analysis of dynamic flow network data.

Conclusions:

• The proposed generative process offers a significant reduction in modeling burden for Bayesian analysis in varying-dimensional spaces.
• This approach bridges the gap between Bayesian uncertainty quantification and frequentist regularization methods.
• The framework is theoretically sound and computationally convenient, with practical utility demonstrated in real-world data analysis.