Asymptotically minimax empirical Bayes estimation of a sparse normal mean vector

By Ryan Martin, Stephen G. Walker. 2013

65 citations recorded by the source corpus.

For the important classical problem of inference on a sparse high-dimensional normal mean vector, we propose a novel empirical Bayes model that admits a posterior distribution with desirable properties under mild conditions. In particular, our empirical Bayes posterior distribution concentrates on balls, centered at the true mean vector, with squared radius proportional to the minimax rate, and its posterior mean is an asymptotically minimax estimator. We also show that, asymptotically, the support of our empirical Bayes posterior has roughly the same effective dimension as the true sparse mean vector. Simulation from our empirical Bayes posterior is straightforward, and our numerical results demonstrate the quality of our method compared to others having similar large-sample properties.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/24dae9aea990b596b0f13b99dd7f4f8cd5b4ef44

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Citations in the corpus

Generalized Fiducial Inference: A Review and New Results (2016, background): “Some argued (Welch and Peers 1963; Martin and Walker 2014) that data-dependent priors are essential in achieving superior frequentist properties in complex statistical problems.”
Uncertainty Quantification for the Horseshoe (with Discussion) (2016, methodology): “For comparison, Figure 1 shows the naive equal-tailed 95% marginal empirical Bayes credible intervals from the same example but based on the two-groups empirical Bayes formulation in Martin and Walker (2014) and our simple Gibbs sampler.”
Bayesian fractional posteriors (2016, background): “from fractional Bayes factors in objective Bayesian model selection [48], data-dependent priors for sparse estimation [41, 42], to marginal likelihood approximation [19] and posterior simulation [20].”
The Horseshoe Estimator: Posterior Concentration around Nearly Black Vectors (2014, methodology): “It is worthy of note that Koenker (2014) generated data according to the same scheme and applied the empirical Bayes procedures due to Martin and Walker (2014) (EBMW) and Koenker and Mizera (2014) (EBKM) to it.”
The Spike-and-Slab LASSO (2018, unknown): “For more background on rates of posterior concentration in regression settings, we refer the reader to Castillo, Schmidt-Hieber, and van der Vaart (2015); Martin and Walker (2014); van der Pas, Kleijn, and van der Vaart (2014); Bhattacharya et al. (2015); and Rockova (2015).”