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Tractable Bayesian variable selection: beyond normality.

David Rossell1, Francisco J Rubio2

  • 1Universitat Pompeu Fabra, Department of Business and Economics, Barcelona (Spain).

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Bayesian variable selection models can be improved by accounting for non-normal data distributions. A new method handles skewness and heavy tails, enhancing accuracy and sensitivity in variable selection.

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Area of Science:

  • Statistical modeling
  • Bayesian inference
  • High-dimensional data analysis

Background:

  • Bayesian variable selection commonly assumes normal distributions, but the impact of misspecification is unclear.
  • Standard methods offer interpretability and convenience but lack flexibility for non-normal data.
  • Existing flexible frameworks may sacrifice analytical tractability.

Purpose of the Study:

  • To propose a tractable extension to Bayesian variable selection that accommodates non-normal data distributions (skewness, heavy tails).
  • To analyze the asymptotic properties of parameter estimation and Bayes factor rates under model misspecification.
  • To investigate strategies for improving inference in the presence of model misspecification.

Main Methods:

  • Developed a log-concave likelihood extension for Bayesian variable selection to handle non-normal data.
  • Characterized asymptotic rates for parameter estimation and Bayes factors under misspecification.
  • Investigated the use of non-local priors for enhanced sparsity and finite-sample performance.
  • Utilized the R package 'mombf' for implementation.

Main Results:

  • The proposed method preserves tractability and facilitates interpretation with non-normal data.
  • Under misspecification, Bayes factors can induce sparsity at similar rates as correct models.
  • Detection rates for signal changes can decrease exponentially under misspecification, reducing sensitivity.
  • Inferring the error distribution can significantly improve inference and ameliorate deficiencies.

Conclusions:

  • Considering the likelihood, not just the prior, is crucial for robust Bayesian variable selection.
  • The proposed extension offers a practical approach to handle non-normal data in variable selection.
  • Inferring error distributions is a valuable strategy to enhance the performance of Bayesian variable selection methods.