Bayes and empirical-Bayes multiplicity adjustment in the variable-selection problem

By James G. Scott, James O. Berger. 2010

615 citations recorded by the source corpus.

This paper studies the multiplicity-correction effect of standard Bayesian variable-selection priors in linear regression. Our first goal is to clarify when, and how, multiplicity correction happens automatically in Bayesian analysis, and to distinguish this correction from the Bayesian Ockham's-razor effect. Our second goal is to contrast empirical-Bayes and fully Bayesian approaches to variable selection through examples, theoretical results and simulations. Considerable differences between the two approaches are found. In particular, we prove a theorem that characterizes a surprising aymptotic discrepancy between fully Bayes and empirical Bayes. This discrepancy arises from a different source than the failure to account for hyperparameter uncertainty in the empirical-Bayes estimate. Indeed, even at the extreme, when the empirical-Bayes estimate converges asymptotically to the true variable-inclusion probability, the potential for a serious difference remains.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/43a4585394e4cb50c365483291a8778b9b29dbda

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Citations in the corpus

Mixtures of g-Priors in Generalized Linear Models (2015, background): “This phenomenon can be explained by the symmetric U-shaped density curve of the Beta-Binomial prior (Scott and Berger 2010, Fig.”
Dirichlet-Laplace priors for optimal shrinkage (2014, background): “In a fully Bayesian framework, it is common to place a beta prior on π , leading to a beta-Bernoulli prior on the model size, which conveys an automatic multiplicity adjustment (Scott and Berger 2010).”
Causal network inference using biochemical kinetics (2014, positive): “Bayesian variable selection requires multiplicity correction to control the false discovery rate and avoid degeneracy (Scott and Berger, 2010).”
Bayesian Subset Modeling for High-Dimensional Generalized Linear Models (2013, background): “Scott and Berger (2010) showed that placing a prior on νn results in an increasing penalty for adding an extra predictor as Pn increases; that is, the fully Bayesian approach provides an automatic correction for the multiplicity involved in variable selection.”
The Median Probability Model and Correlated Variables (2018, background): “Beta-binomial Prior: It is generally acknowledged (Cui and George 2008; Ley and Steel 2009; Scott and Berger 2010) that assigning equal prior probability to all models is a poor choice, since it does not adjust for the multiple testing that is effectively being done in variable selection.”
Criteria for Bayesian model choice with application to variable selection (2012, background): “The prior (8) has its origins in the robust prior introduced by Strawderman (1971) and Berger (1980, 1985), for estimating a k-variate normal mean β in the sampling scheme β̂ ∼Nk(β,Σ).”
BAYESIAN LINEAR REGRESSION WITH SPARSE PRIORS (2014, background): “2) were considered by many authors, including [8, 20, 21, 26, 34, 41, 48].”
Local shrinkage rules, Lévy processes and regularized regression (2010, methodology): “This is the typical assumption that is used in Bayesian model selection, model averaging and multiple testing (see George and Foster (2000) and Scott and Berger (2010)).”
A Default Bayesian Hypothesis Test for ANOVA Designs (2012, methodology): “To correct for multiple comparisons, the prior on the model itself must be chosen appropriately (see Stephens and Balding 2009; Scott and Berger 2010, and references therein).”
Objective Priors for Discrete Parameter Spaces (2012, background): “Steel (2007) and Scott & Berger (2008) ) that this assignment of prior model probabilities”
Handling Sparsity via the Horseshoe (2009, background): “See Scott and Berger (2008) for an extensive discussion of this phe- nomenon.”
Joint estimation of multiple networks from time course data (2013, background): “There are a number of interesting extensions which may be considered in future work: (i) In high dimensions, Bayesian variable selection requires multiplicity correction in order to avoid degeneracy (Scott and Berger, 2010).”
scCODA is a Bayesian model for compositional single-cell data analysis (2021, background): “As noted previously 33,34 , Bayesian variable selection methods must control for multiplicity, to avoid an in ﬂ ated number of false-positive associations.”
Shrink Globally, Act Locally: Sparse Bayesian Regularization and Prediction (2012, background): “It is by no means certain that the two procedures will reach similar answers asymptotically, since this difference in functional form persists for all p [see, for example, Scott and Berger, 2010].”