In the archetype discovery problem a researcher wants to summarize N heterogeneous policy effects of interest that vary over a discrete set of covariates. The goal is to partition the set of covariates into K<N groups -- the archetype sets -- and to provide a summary of the policy effects for each group. We use decision theory to show that, under a weighted mean-squared-error criterion, a procedure analogous to the Sorted Group Average Treatment Effects (GATES) solves the archetype discovery problem. The key difference is that, in the optimal procedure, archetype sets are obtained by weighted K-means clustering of the N heterogeneous policy effects, instead of relying on K equally-spaced quantiles. We show that the procedure that minimizes average risk for a given prior can be obtained by clustering the different values of the posterior mean estimate of the policy effects of interest. Similarly, an approximately minimax procedure in large samples can be obtained by clustering a consistent estimator of the policy effects. In both of these cases, an exact solution to the weighted K-means clustering problem can be found using a simple and well-known dynamic programming algorithm.

翻译：在原型发现问题中，研究者希望总结N个随离散协变量变化的异质性政策效应。目标是：将协变量集合划分为K<N个组（即原型集），并为每个组提供政策效应的摘要。我们运用决策理论证明，在加权均方误差准则下，一种类似于排序分组平均处理效应（GATES）的方法可解决原型发现问题。关键区别在于，最优方法中，原型集通过N个异质性政策效应的加权K均值聚类获得，而非依赖K个等间距分位数。我们证明：对于给定先验，最小化平均风险的方法可以通过对目标政策效应后验均值估计的不同值进行聚类得到；类似地，大样本下的近似极小极大方法可通过聚类政策效应的一致估计量获得。在这两种情况下，加权K均值聚类问题的精确解均可通过简单且广为人知的动态规划算法求得。