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Learning from a lot: Empirical Bayes for high-dimensional model-based prediction.

Mark A van de Wiel1,2, Dennis E Te Beest1, Magnus M Münch1,3

  • 1Department of Epidemiology and Biostatistics, Amsterdam Public Health Research Institute VU University Medical Center Amsterdam The Netherlands.

Scandinavian Journal of Statistics, Theory and Applications
|April 23, 2019
PubMed
Summary

Empirical Bayes methods offer powerful ways to learn from numerous variables and prior data. These techniques enhance predictions in penalized regression, linear discriminant analysis, and Bayesian models, especially when incorporating "co-data".

Keywords:
co‐dataempirical Bayesmarginal likelihoodpredictionvariable selection

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Area of Science:

  • Statistics
  • Machine Learning
  • Bioinformatics

Background:

  • Empirical Bayes (EB) methods provide a framework for learning from large datasets with many variables and extensive prior information.
  • Prior information can be sourced from public repositories, enhancing predictive modeling.
  • Existing prediction methods like penalized regression and linear discriminant analysis can be improved using EB.

Purpose of the Study:

  • To review and discuss various empirical Bayes methods and their applications in model-based prediction.
  • To explore both formal (marginal likelihood maximization) and informal EB approaches.
  • To investigate the utility of EB, particularly when incorporating multi-parameter priors ('co-data').

Main Methods:

  • Review of empirical Bayes applications in penalized regression, linear discriminant analysis, and Bayesian models (sparse/dense priors).
  • Discussion of formal EB (marginal likelihood) and informal EB (data summaries).
  • Comparison of EB with cross-validation and full Bayes, including hybrid approaches.
  • Analysis of a simple EB estimator in a linear model to understand its performance with varying numbers of variables (p).

Main Results:

  • Empirical Bayes methods are versatile for learning from high-dimensional data and prior information.
  • EB is particularly effective when prior information includes multiple parameters ('co-data').
  • Novel examples demonstrate EB's ability to integrate multiple co-data sources: a Bayesian spike-and-slab model and a hybrid EB-full Bayes ridge regression for predictive intervals.

Conclusions:

  • Empirical Bayes offers a robust approach for enhancing predictive models by leveraging large-scale data and prior knowledge.
  • The incorporation of 'co-data' through advanced EB methods significantly improves estimation and prediction.
  • The presented novel methods offer practical solutions for complex modeling scenarios, including the estimation of posterior predictive intervals.