Sequential Gibbs posteriors with applications to principal component analysis.

Area of Science:

• Statistics
• Bayesian Inference
• Machine Learning

Background:

• Gibbs posteriors offer a Bayesian framework for likelihood-free inference.
• A single tuning parameter in Gibbs posteriors often results in poor uncertainty quantification and inadequate frequentist coverage.
• Existing methods struggle with accurate credible region coverage, even in large sample sizes.

Purpose of the Study:

• To propose a sequential extension of Gibbs posteriors to enhance uncertainty quantification.
• To address the limitations of traditional Gibbs posteriors in achieving nominal frequentist coverage.
• To provide a principled Bayesian inference method with improved uncertainty estimates.

Main Methods:

• Developed a sequential extension to Gibbs posteriors.
• Proved concentration properties for the sequential posterior.
• Established a Bernstein-von Mises theorem for the sequential posterior on Euclidean spaces and manifolds.
• Derived the first Bernstein-von Mises theorem for likelihood-based Bayesian posteriors on manifolds.

Main Results:

• The sequential Gibbs posterior demonstrates concentration properties.
• The sequential posterior satisfies a Bernstein-von Mises theorem under verifiable conditions.
• The study provides the first Bernstein-von Mises theorem for Bayesian posteriors on manifolds.
• The proposed methods were successfully applied to principal component analysis.

Conclusions:

• Sequential Gibbs posteriors offer a principled and effective solution for accurate uncertainty quantification in Bayesian inference.
• The theoretical guarantees (concentration and Bernstein-von Mises theorem) validate the proposed method.
• This work extends Bayesian inference theory to manifolds and improves upon traditional likelihood-free methods.