Uncertainty Quantification for the Horseshoe (with Discussion)

We begin by introducing the main ideas of the paper under discussion. We discuss some interesting issues regarding adaptive component-wise credible intervals. We then briefly touch upon the concepts of self-similarity and excessive bias restriction. This is then followed by some comments on the extensive simulation study carried out in the paper.

[1]  D. Fraser,et al.  Bayes, Reproducibility and the Quest for Truth , 2016 .

[2]  A. W. van der Vaart,et al.  Adaptive Bayesian credible bands in regression with a Gaussian process prior , 2015, Sankhya A.

[3]  Prasenjit Ghosh,et al.  Asymptotic Optimality of One-Group Shrinkage Priors in Sparse High-dimensional Problems , 2017 .

[4]  Arnaud Doucet,et al.  Sparse Bayesian nonparametric regression , 2008, ICML '08.

[5]  A. V. D. Vaart,et al.  BAYESIAN LINEAR REGRESSION WITH SPARSE PRIORS , 2014, 1403.0735.

[6]  D. Picard,et al.  Adaptive confidence interval for pointwise curve estimation , 2000 .

[7]  D. A. S. Fraser Is Bayes Posterior just Quick and Dirty Confidence , 2011 .

[8]  N. Pillai,et al.  Dirichlet–Laplace Priors for Optimal Shrinkage , 2014, Journal of the American Statistical Association.

[9]  R. Tibshirani Regression Shrinkage and Selection via the Lasso , 1996 .

[10]  Sara van de Geer,et al.  Statistics for High-Dimensional Data , 2011 .

[11]  J. Griffin,et al.  Inference with normal-gamma prior distributions in regression problems , 2010 .

[12]  Michael I. Jordan,et al.  Dimensionality Reduction for Supervised Learning with Reproducing Kernel Hilbert Spaces , 2004, J. Mach. Learn. Res..

[13]  R. Nickl,et al.  A sharp adaptive confidence ball for self-similar functions , 2014, 1406.3994.

[14]  Van Der Vaart,et al.  The Horseshoe Estimator: Posterior Concentration around Nearly Black Vectors , 2014, 1404.0202.

[15]  Nicholas G. Polson,et al.  The Horseshoe+ Estimator of Ultra-Sparse Signals , 2015, 1502.00560.

[16]  Cun-Hui Zhang,et al.  Confidence intervals for low dimensional parameters in high dimensional linear models , 2011, 1110.2563.

[17]  Malay Ghosh,et al.  Asymptotic Properties of Bayes Risk of a General Class of Shrinkage Priors in Multiple Hypothesis Testing Under Sparsity , 2013, 1310.7462.

[18]  Adam D. Bull,et al.  Honest adaptive confidence bands and self-similar functions , 2011, 1110.4985.

[19]  Bin Yu,et al.  Asymptotic Properties of Lasso+mLS and Lasso+Ridge in Sparse High-dimensional Linear Regression , 2013, 1306.5505.

[20]  Wenhua Jiang,et al.  General maximum likelihood empirical Bayes estimation of normal means , 2009, 0908.1709.

[21]  S. Geer,et al.  On asymptotically optimal confidence regions and tests for high-dimensional models , 2013, 1303.0518.

[22]  Enes Makalic,et al.  A Simple Sampler for the Horseshoe Estimator , 2015, IEEE Signal Processing Letters.

[23]  Harry van Zanten,et al.  Honest Bayesian confidence sets for the L2-norm , 2013, 1311.7474.

[24]  Jayanta K. Ghosh,et al.  Asymptotic Properties of Bayes Risk for the Horseshoe Prior , 2013 .

[25]  Stephen G. Walker,et al.  Empirical Bayes posterior concentration in sparse high-dimensional linear models , 2014, 1406.7718.

[26]  Stephen G. Walker,et al.  Asymptotically minimax empirical Bayes estimation of a sparse normal mean vector , 2013, 1304.7366.

[27]  I. Johnstone,et al.  Needles and straw in haystacks: Empirical Bayes estimates of possibly sparse sequences , 2004, math/0410088.

[28]  James G. Scott,et al.  Shrink Globally, Act Locally: Sparse Bayesian Regularization and Prediction , 2022 .

[29]  James G. Scott,et al.  On the half-cauchy prior for a global scale parameter , 2011, 1104.4937.

[30]  Prasenjit Ghosh,et al.  Posterior Concentration Properties of a General Class of Shrinkage Priors around Nearly Black Vectors , 2014 .

[31]  V. Johnson,et al.  On the use of non‐local prior densities in Bayesian hypothesis tests , 2010 .

[32]  James G. Scott,et al.  The horseshoe estimator for sparse signals , 2010 .

[33]  Aad van der Vaart,et al.  Adaptive posterior contraction rates for the horseshoe , 2017, 1702.03698.

[34]  S. Geer,et al.  The adaptive and the thresholded Lasso for potentially misspecified models (and a lower bound for the Lasso) , 2011 .

[35]  Eduard Belitser,et al.  Needles and straw in a haystack: robust empirical Bayes confidence for possibly sparse sequences , 2015 .

[36]  Jiunn T. Hwang,et al.  The Nonexistence of 100$(1 - \alpha)$% Confidence Sets of Finite Expected Diameter in Errors-in-Variables and Related Models , 1987 .

[37]  Sara van de Geer,et al.  Confidence sets in sparse regression , 2012, 1209.1508.

[38]  A. W. Vaart,et al.  Frequentist coverage of adaptive nonparametric Bayesian credible sets , 2013, 1310.4489.

[39]  Ker-Chau Li,et al.  Honest Confidence Regions for Nonparametric Regression , 1989 .

[40]  Tatyana Krivobokova,et al.  Adaptive empirical Bayesian smoothing splines , 2014, 1411.6860.

[41]  J. Robins,et al.  Adaptive nonparametric confidence sets , 2006, math/0605473.

[42]  James G. Scott Bayesian Estimation of Intensity Surfaces on the Sphere via Needlet Shrinkage and Selection , 2011 .

[43]  James G. Scott,et al.  Good, great, or lucky? Screening for firms with sustained superior performance using heavy-tailed priors , 2010, 1010.5223.

[44]  James G. Scott Parameter expansion in local-shrinkage models , 2010, 1010.5265.

[45]  R. Nickl,et al.  On the Bernstein–von Mises phenomenon for nonparametric Bayes procedures , 2013, 1310.2484.

[46]  R. Nickl,et al.  CONFIDENCE BANDS IN DENSITY ESTIMATION , 2010, 1002.4801.

[47]  Jaeyong Lee,et al.  GENERALIZED DOUBLE PARETO SHRINKAGE. , 2011, Statistica Sinica.

[48]  A. V. D. Vaart,et al.  Needles and Straw in a Haystack: Posterior concentration for possibly sparse sequences , 2012, 1211.1197.

[49]  J. Rousseau,et al.  Asymptotic frequentist coverage properties of Bayesian credible sets for sieve priors , 2016, The Annals of Statistics.

[50]  Eduard Belitser,et al.  Needles and straw in a haystack: Robust confidence for possibly sparse sequences , 2015, Bernoulli.

[51]  A. V. D. Vaart,et al.  Credible sets in the fixed design model with Brownian motion prior , 2015 .

[52]  James G. Scott,et al.  Handling Sparsity via the Horseshoe , 2009, AISTATS.

[53]  Kolyan Ray Adaptive Bernstein–von Mises theorems in Gaussian white noise , 2014, 1407.3397.

[54]  Empirical priors for targeted posterior concentration rates , 2016, 1604.05734.