The beta distribution serves as a canonical tool for modeling probabilities and is extensively used in statistics and machine learning, especially in the field of Bayesian nonparametrics. Despite its widespread use, there is limited work on flexible and computationally convenient stochastic process extensions for modeling dependent random probabilities. We propose a novel stochastic process called the logistic-beta process, whose logistic transformation yields a stochastic process with common beta marginals. Similar to the Gaussian process, the logistic-beta process can model dependence on both discrete and continuous domains, such as space or time, and has a highly flexible dependence structure through correlation kernels. Moreover, its normal variance-mean mixture representation leads to highly effective posterior inference algorithms. The flexibility and computational benefits of logistic-beta processes are demonstrated through nonparametric binary regression simulation studies. Furthermore, we apply the logistic-beta process in modeling dependent Dirichlet processes, and illustrate its application and benefits through Bayesian density regression problems in a toxicology study.

翻译：Beta分布作为概率建模的标准工具，广泛应用于统计学与机器学习领域，尤其在贝叶斯非参数方法中占据重要地位。尽管应用广泛，但目前关于构建灵活且计算便捷的随机过程扩展以建模相关随机概率的研究仍十分有限。本文提出了一种名为Logistic-β过程的新型随机过程，其Logistic变换可生成具有共同Beta边缘分布的随机过程。与高斯过程类似，Logistic-β过程能够对离散域与连续域（如空间或时间）上的依赖性进行建模，并通过相关核实现高度灵活的依赖结构。此外，其正态方差-均值混合表示形式可推导出高效的后验推断算法。通过非参数二元回归仿真实验，验证了Logistic-β过程的灵活性与计算优势。进一步地，我们将Logistic-β过程应用于相关Dirichlet过程的建模，并在毒理学研究的贝叶斯密度回归问题中展示了其应用价值与优势。