Modern regression applications can involve hundreds or thousands of variables which motivates the use of variable selection methods. Bayesian variable selection defines a posterior distribution on the possible subsets of the variables (which are usually termed models) to express uncertainty about which variables are strongly linked to the response. This can be used to provide Bayesian model averaged predictions or inference, and to understand the relative importance of different variables. However, there has been little work on meaningful representations of this uncertainty beyond first order summaries. We introduce Cartesian credible sets to address this gap. The elements of these sets are formed by concatenating sub-models defined on each block of a partition of the variables. Investigating these sub-models allow us to understand whether the models in the Cartesian credible set always/never/sometimes include a particular variable or group of variables and provide a useful summary of model uncertainty. We introduce methods to find these sets that emphasize ease of understanding. The potential of the method is illustrated on regression problems with both small and large numbers of variables.

翻译：现代回归应用可能涉及成百上千个变量，这促使了变量选择方法的使用。贝叶斯变量选择在变量的可能子集（通常称为模型）上定义后验分布，以表达哪些变量与响应变量强相关的不确定性。这可用于提供贝叶斯模型平均预测或推断，并理解不同变量的相对重要性。然而，除了一阶汇总外，关于如何有意义地表示这种不确定性的工作较少。我们引入笛卡尔可信集以填补这一空白。这些集合的元素由变量划分的每个区块上定义的子模型连接而成。研究这些子模型能够帮助我们理解笛卡尔可信集中的模型是否总是/从不/有时包含某个特定变量或变量组，并提供模型不确定性的有用汇总。我们引入了强调易理解性的方法来寻找这些集合。该方法的潜力通过包含少量和大量变量的回归问题得以展示。