Asymptotic Properties of Bayes Risk for the Horseshoe Prior

By Jyotishka Datta, Jayanta Kumar Ghosh. 2013

114 citations recorded by the source corpus.

In this paper, we establish some optimality properties of the multiple testing rule induced by the horseshoe estimator due to Carvalho, Polson, and Scott (2010, 2009) from a Bayesian decision theoretic viewpoint. We consider the two- groups model for the data and an additive loss structure such that the total loss is equal to the number of misclassified hypotheses. We use the same asymptotic framework as Bogdan, Chakrabarti, Frommlet, and Ghosh (2011) who introduced the Bayes oracle in the context of multiple testing and provided conditions under which the Benjamini-Hochberg and Bonferroni procedures attain the risk of the Bayes oracle. We prove a similar result for the horseshoe decision rule up to Op1q with the constant in the horseshoe risk close to the constant in the oracle. We use the Full Bayes estimate of the tuning parameter . It is worth noting that the Full Bayes estimate cannot be replaced by the Empirical Bayes estimate, which tends to be too small.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b80d03df15dd222f1652a09c1696269c7d0c34ff

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Citations in the corpus

Uncertainty Quantification for the Horseshoe (with Discussion) (2016, methodology): “We refer to [14] for theoretical properties of this procedure, and compare the credible interval and thresholding methods further through simulation in Section 5.”
Adaptive posterior contraction rates for the horseshoe (2017, methodology): “We found the observation in [10] that an empirical Bayes approach cannot replace a hierarchical Bayes one, because the estimate of τ tends to be too small, too general.”
Sparsity information and regularization in the horseshoe and other shrinkage priors (2017, background): “As discussed in the introduction, the horseshoe prior has been shown to possess several desirable theoretical properties and good performance in practice (Carvalho, Polson and Scott, 2009, 2010; Polson and Scott, 2011; Datta and Ghosh, 2013; van der Pas, Kleijn and van der Vaart, 2014).”
The Graphical Horseshoe Estimator for Inverse Covariance Matrices (2017, methodology): “These include improved Kullback–Leibler risk bounds (Carvalho, Polson, and Scott 2010), asymptotic optimality in testing under 0–1 loss (Datta and Ghosh 2013), minimaxity in estimation under the 2 loss (van der Pas, Kleijn, and van der Vaart 2014), and improved risk properties in linear regression…”
Asymptotic Properties of Bayes Risk of a General Class of Shrinkage Priors in Multiple Hypothesis Testing Under Sparsity (2013, methodology): “Note that the above estimator of τ is truncated below by 1m and hence it is not susceptible to collapsing to zero, which is a major concern for the use of such empirical Bayes approaches as mentioned in Carvalho et al. (2009), Scott and Berger (2010), Bogdan et al. (2008) and Datta and Ghosh (2013).”
Comparison of Bayesian predictive methods for model selection (2015, background): “, 2009, 2010) has been demonstated to have desirable properties empirically and theoretically assuming a properly chosen shrinkage factor (Datta and Ghosh, 2013; van der Pas et al., 2014).”