arXiv 22 Jun 2026 · Statistics — Methodology
arXiv:2606.23509 · PDF · DOI · OpenAlex · Extracted main text
We study how the choice of default prior for a common Gaussian scale affects high-dimensional shrinkage risk, highlighting the role played by high-dimensional geometry. Formally, we consider a high-dimensional setting in which the near-zero behavior of the common scale prior has first-order consequences for shrinkage risk, and show that priors that are flat on the variance and those flat on the standard deviation allocate markedly different mass near the zero-scale boundary, leading to distinct shrinkage behavior and informing principled default prior selection. Specifically, under a radial-power benchmark, we establish that the SD-flat benchmark has a one-unit asymptotic risk advantage near the origin, crosses over in the critical regime, and is second-order equivalent to the variance-flat benchmark for strong signals. Proper single global-scale hyperpriors and bounded coordinate-multiplier mixtures inherit these limits through the near-zero exponent of their SD-scale density. For heavier-tailed or sparse priors, that exponent still classifies the common global-scale component, while local-scale tails, model-size priors, or allocation priors can also affect risk.
appendix boundary found by appendix_command · 36% of the source is main text. Read the extracted text to check this.
The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Gelman, A (2006) Prior distributions for variance parameters in hierarchical models | 0.928 | 4 | 3 | 100% |
| 2 | Polson, N. G. and J. G. Scott (2012) On the Half-Cauchy Prior for a Global Scale Parameter | 0.928 | 4 | 3 | 100% |
| 3 | Brown, L. D (1971) Admissible estimators, recurrent diffusions, and insoluble boundary value problems | 0.737 | 3 | 2 | 100% |
| 4 | Strawderman, W. E (1971) Proper Bayes minimax estimators of the multivariate normal mean | 0.737 | 3 | 2 | 100% |
| 5 | Brown, L. D. and L. H. Zhao (2012) A geometrical explanation of Stein shrinkage | 0.644 | 2 | 2 | 100% |
| 6 | Carvalho, C. M., N. G. Polson, and J. G. Scott (2010) The horseshoe estimator for sparse signals | 0.644 | 2 | 2 | 100% |
| 7 | Johnstone, I. M. and B. W. Silverman (2004) Needles and straw in haystacks: empirical Bayes estimates of possibly sparse sequences | 0.644 | 2 | 2 | 100% |
| 8 | Ledoux, M (2001) The Concentration of Measure Phenomenon | 0.644 | 2 | 2 | 100% |
| 9 | Maruyama, Y. and A. Takemura (2008) Admissibility and minimaxity of generalized Bayes estimators for spherically symmetric family | 0.644 | 2 | 2 | 100% |
| 10 | Piironen, J. and A. Vehtari (2017) a): On the Hyperprior Choice for the Global Shrinkage Parameter in the Horseshoe Prior, in | 0.644 | 2 | 2 | 100% |
Showing the top 10 of 31 scored citations.