Scalable Empirical Bayes Inference and Bayesian Sensitivity Analysis

Area of Science:

• Statistics
• Computational Statistics
• Bayesian Inference

Background:

• Bayesian analysis involves observing data $$ dependent on parameter $$ , with an unknown prior distribution $$ .
• Subjective Bayesian methods struggle with determining the precise prior, while empirical Bayes estimates the latent distribution from data.
• Common empirical Bayes methods maximize marginal likelihood $$ , but analytic evaluation is often infeasible and existing procedures may be inaccurate or scale poorly.

Purpose of the Study:

• To review and critique existing literature on estimating the latent distribution in empirical Bayes.
• To introduce a novel, general, and dimensionally scalable method for estimating the latent distribution using Markov chain Monte Carlo (MCMC).
• To demonstrate the utility of the proposed method for obtaining point estimates and confidence bands for the prior family.

Main Methods:

• Literature review of current empirical Bayes estimation techniques.
• Development of a new estimation method based on Markov chain Monte Carlo (MCMC).
• Application of the MCMC method to derive posterior expectations and confidence bands.

Main Results:

• Existing empirical Bayes estimation methods are found to be either inaccurate or computationally inefficient for high-dimensional problems.
• The proposed MCMC-based method provides a generally applicable and scalable solution for estimating the latent distribution.
• The methodology facilitates the computation of point estimates and globally-valid confidence bands for the prior family.

Conclusions:

• The developed MCMC approach offers a significant improvement over traditional methods for empirical Bayes estimation.
• The method's generality and scalability make it suitable for a wide range of statistical problems.
• The approach provides valuable tools for characterizing uncertainty in the estimated prior distributions.