Carvalho (2010) established two foundational theorems for the horseshoe prior: tight two-sided logarithmic bounds on the marginal density near the origin (Theorem~1.1), and a super-efficient rate of convergence of the Bayes predictive density to the true sampling density in sparse situations (Theorem~2). The ``Shrink Globally, Act Locally'' paper \citep{polson2010shrink} formalised necessary and sufficient conditions on the prior's behaviour at the origin for sparsity adaptation as $p \to \infty$. We show that these results are not merely descriptive properties of the horseshoe -- they are the finite-sample precursors to the asymptotic moderate deviation principle (MDP) of \citet{datta2026newlook}. The log-pole singularity $\piH(θ) \asymp -\log\absθ$ is precisely the origin integrability boundary that selects the MDP threshold $\tcrit = \sqrt{\log(πn/2)}$; super-efficiency below the threshold and tail robustness above it together produce the ABOS Bayes risk $p_0 \log(p/p_0)/n$; and the Clarke--Barron information-theoretic asymptotics of Bayes methods provide the unifying framework in which all three results are faces of a single logarithmic budget principle.

翻译：Carvalho(2010)为马蹄先验建立了两个基础定理：原点附近边际密度紧的双侧对数界(定理1.1)，以及稀疏情形下贝叶斯预测密度向真实抽样密度的超高效收敛速率(定理2)。《全局收缩，局部行动》论文\citep{polson2010shrink}形式化了当$p \to \infty$时先验在原点处行为实现稀疏自适应的充要条件。我们证明这些结果并非仅是马蹄先验的描述性性质——它们是\citet{datta2026newlook}渐近中偏差原理(MDP)的有限样本前驱。对数极点奇异性$\piH(θ) \asymp -\log\absθ$正是选择MDP阈值$\tcrit = \sqrt{\log(πn/2)}$的原点可积性边界；阈值以下的超高效与阈值以上的尾部稳健性共同产生ABOS贝叶斯风险$p_0 \log(p/p_0)/n$；而贝叶斯方法的Clarke-Barron信息论渐近性提供了统一框架，使这三个结果成为单一对数预算原理的不同侧面。