Predictive inference in the sparse Gaussian sequence model has received considerably less attention than its non-sparse, finite-sample counterpart. Existing work has largely been confined to discrete mixture priors. In this paper, we study predictive inference under a widely used continuous mixture prior, the Horseshoe. We provide new theoretical results establishing exact asymptotic minimax optimality of the predictive Bayes estimator when the sparsity level is known. Furthermore, through a Gaussian-mixture representation of the posterior predictive density (which we term Horseshoe spectroscopy), the phase-transition in the local shrinkage scale is inherited by the predictive mechanism, producing behavior similar to that of previous thresholding/switching estimators. When sparsity is unknown, we adopt a fully Bayesian approach using a hierarchical Horseshoe prior and show that it performs adaptive, as opposed to manual, switching. Under a theta-min condition, the resulting predictive risk admits an upper bound over a restricted parameter class that is sharper than the minimax rate over the full class. We demonstrate the practical value of predictive Horseshoe shrinkage on data such as images and time series that can be naturally modeled as sparse Gaussian sequences. We illustrate this approach on facial recognition across varying facial expressions and study region-wise atypical brain lateralization in autism spectrum disorder.

翻译：在稀疏高斯序列模型中，预测推断受到的关注远少于其非稀疏有限样本对应物。现有研究主要局限于离散混合先验。本文研究了广泛使用的连续混合先验——马靴先验下的预测推断。当稀疏水平已知时，我们提供了新的理论结果，建立了预测贝叶斯估计量的精确渐近极小化最优性。此外，通过后验预测密度的高斯混合表示（我们称之为马靴谱分析），局部收缩尺度的相变特性被预测机制继承，产生类似于先前阈值/切换估计量的行为。当稀疏性未知时，我们采用全贝叶斯方法，使用分层马靴先验，并证明其执行自适应切换而非手动切换。在theta-min条件下，所得预测风险在受限参数类上的上界比全参数类上的极小化风险率更紧。我们通过可自然建模为稀疏高斯序列的图像和时间序列等数据，展示了预测性马靴收缩的实际价值。我们将该方法应用于跨不同面部表情的人脸识别，并研究了自闭症谱系障碍中区域特异性脑偏侧化。