This article investigates Bayesian variable selection when there is a hierarchical dependence structure on the inclusion of predictors in the model. In particular, we study the type of dependence found in polynomial response surfaces of orders two and higher, whose model spaces are required to satisfy weak or strong heredity conditions. These conditions restrict the inclusion of higher-order terms depending upon the inclusion of lower-order parent terms. We develop classes of priors on the model space, investigate their theoretical and finite sample properties, and provide a Metropolis–Hastings algorithm for searching the space of models. The tools proposed allow fast and thorough exploration of model spaces that account for hierarchical polynomial structure in the predictors and provide control of the inclusion of false positives in high posterior probability models.

中文翻译：

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受遗传条件约束的模型空间的贝叶斯变量选择

本文研究了当模型中包含的预测变量存在分层依赖结构时的贝叶斯变量选择。特别是，我们研究了在二阶及更高阶多项式响应曲面中发现的依赖类型，其模型空间需要满足弱或强遗传条件。这些条件根据包含的低阶父项来限制高阶项的包含。我们在模型空间上开发先验类别，研究它们的理论和有限样本属性，并提供用于搜索模型空间的 Metropolis-Hastings 算法。