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数百或数千个变量，这推动了变量选择方法的使用。贝叶斯变量选择在可能的变量子集（通常称为模型）上定义一个后验分布，以表达关于哪些变量与响应强相关的不确定性。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，除了其一阶摘要之外，目前对此类不确定性的有意义的表示方法研究较少。我们引入笛卡尔可信集来填补这一空白。这些集合的元素是通过连接变量划分的每个块上定义的子模型而构建的。研究这些子模型使我们能够理解笛卡尔可信集中的模型是总是/从不/有时包含特定变量或变量组，从而提供模型不确定性的有用摘要。我们提出了强调易于理解的方法来寻找这些集合。本文通过变量数量较少和较多的回归问题阐明了该方法的潜力。