Bayesian Linear Inverse Problems in Regularity Scales with Discrete Observations

Area of Science:

• Bayesian inference
• Inverse problems
• Statistical learning theory

Background:

• Posterior distributions quantify uncertainty in inverse problems.
• Understanding contraction rates is crucial for reliable estimation.
• General methods are needed for diverse prior choices.

Purpose of the Study:

• Derive abstract contraction rates for posterior distributions.
• Analyze the impact of prior properties on estimation accuracy.
• Evaluate performance of specific prior types in inverse problems.

Main Methods:

• Develop abstract theoretical results for general priors.
• Analyze contraction rates based on discrete Galerkin approximation.
• Investigate prior concentration and approximation properties.

Main Results:

• Contraction rates are determined by discrete approximation quality.
• Prior concentration near the true solution is key.
• Non-conjugate series, Gaussian, and mixture priors achieve near-optimal, adaptive recovery.

Conclusions:

• The derived abstract results provide a general framework for analyzing posterior contraction rates.
• Specific prior choices, like mixtures of Gaussians, demonstrate strong performance.
• The findings advance the understanding of Bayesian inference in inverse problems.