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Summary
This summary is machine-generated.

We derive contraction rates for posterior distributions in inverse problems using discrete observations. These rates depend on prior concentration and approximation quality, enabling near-optimal recovery with various priors.

Keywords:
Adaptive estimationFixed designGalerkinGaussian priorHilbert scaleInterpolationLinear inverse problemNonparametric Bayesian estimationPosterior contraction rateRandom series priorRegressionRegularity scale

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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.