Many statistical analyses, in both observational data and randomized control trials, ask: how does the outcome of interest vary with combinations of observable covariates? How do various drug combinations affect health outcomes, or how does technology adoption depend on incentives and demographics? Our goal is to partition this factorial space into "pools" of covariate combinations where the outcome differs across the pools (but not within a pool). Existing approaches (i) search for a single "optimal" partition under assumptions about the association between covariates or (ii) sample from the entire set of possible partitions. Both these approaches ignore the reality that, especially with correlation structure in covariates, many ways to partition the covariate space may be statistically indistinguishable, despite very different implications for policy or science. We develop an alternative perspective, called Rashomon Partition Sets (RPSs). Each item in the RPS partitions the space of covariates using a tree-like geometry. RPSs incorporate all partitions that have posterior values near the maximum a posteriori partition, even if they offer substantively different explanations, and do so using a prior that makes no assumptions about associations between covariates. This prior is the $\ell_0$ prior, which we show is minimax optimal. Given the RPS we calculate the posterior of any measurable function of the feature effects vector on outcomes, conditional on being in the RPS. We also characterize approximation error relative to the entire posterior and provide bounds on the size of the RPS. Simulations demonstrate this framework allows for robust conclusions relative to conventional regularization techniques. We apply our method to three empirical settings: price effects on charitable giving, chromosomal structure (telomere length), and the introduction of microfinance.

翻译：在许多统计分析中，无论是观测数据还是随机对照试验，都会提出这样的问题：感兴趣的结果如何随可观测协变量组合的变化而变化？各种药物组合如何影响健康结果，或者技术采纳如何取决于激励措施和人口统计特征？我们的目标是将该因子空间划分为协变量组合的“池”，使得结果在不同池之间存在差异（但在同一池内无差异）。现有方法（i）在关于协变量间关联的假设下搜索单一“最优”划分，或（ii）从所有可能划分的集合中进行抽样。这两种方法都忽略了一个现实：尤其在协变量存在相关结构的情况下，许多划分协变量空间的方式可能在统计上无法区分，尽管它们对政策或科学的意义截然不同。我们提出了一种称为Rashomon划分集（RPSs）的替代视角。RPS中的每个项目使用树状几何划分协变量空间。RPS包含所有后验值接近最大后验划分的划分，即使它们提供了实质上不同的解释，并且使用一个不对协变量间关联做任何假设的先验来实现这一点。该先验是$\ell_0$先验，我们证明其具有极小极大最优性。给定RPS，我们计算特征效应向量对结果的任何可测函数的后验，条件是该函数属于RPS。我们还刻画了相对于整个后验的近似误差，并提供了RPS大小的界限。模拟实验表明，相对于传统正则化技术，该框架允许得出更稳健的结论。我们将该方法应用于三个实证场景：价格对慈善捐赠的影响、染色体结构（端粒长度）以及小额信贷的引入。