Isotonic regression or monotone function estimation is a problem of estimating function values under monotonicity constraints, which appears naturally in many scientific fields. This paper proposes a new Bayesian method with global-local shrinkage priors for estimating monotone function values. Specifically, we introduce half shrinkage priors for positive valued random variables and assign them for the first-order differences of function values. We also develop fast and simple Gibbs sampling algorithms for full posterior analysis. By incorporating advanced shrinkage priors, the proposed method is adaptive to local abrupt changes or jumps in target functions. We show this adaptive property theoretically by proving that the posterior mean estimators are robust to large differences and that asymptotic risk for unchanged points can be improved. Finally, we demonstrate the proposed methods through simulations and applications to a real data set.

翻译：保序回归（即单调函数估计）是在单调性约束下估计函数值的问题，该问题自然存在于众多科学领域中。本文提出一种融合全局-局部收缩先验的贝叶斯新方法，用于估计单调函数值。具体而言，我们针对正随机变量引入半收缩先验，并将这些先验赋予函数值的一阶差分项。同时，我们开发了快速简洁的吉布斯采样算法以实现完整的后验分析。通过整合先进收缩先验，所提方法能够自适应目标函数中的局部突变或跳跃。我们从理论上证明了这种自适应特性：后验均值估计量对较大差异具有稳健性，且可提升非突变点的渐近风险。最后，通过模拟实验与真实数据集的应用验证了所提方法的有效性。