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 a method to find these sets that emphasizes ease of understanding and can be easily computed from Markov chain Monte Carlo output. 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甚至数千个变量，这促使变量选择方法的使用。贝叶斯变量选择通过对变量可能子集（通常称为模型）定义后验分布，来表达哪些变量与响应变量存在强关联的不确定性。该方法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，除一阶摘要外，关于这种不确定性的有意义表示的研究尚不充分。为填补这一空白，我们提出了笛卡尔可信集。该集合的元素通过拼接变量划分各区块上定义的子模型构成。通过研究这些子模型，我们可以判断笛卡尔可信集中的模型是否总是/从不/有时包含特定变量或变量组，从而为模型不确定性提供有效的概括性描述。我们提出了一种强调可理解性的构建方法，该方法可直接基于马尔可夫链蒙特卡洛输出进行高效计算。通过在变量规模各异（从少量到大量变量）的回归问题上进行实验，验证了该方法的潜在应用价值。