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：在观察性数据和随机对照试验的许多统计分析中，研究者常追问：结果变量如何随可观测协变量的组合变化？不同药物组合如何影响健康结局？技术采用如何取决于激励因素和人口特征？我们的目标是将这一因子空间划分为若干协变量组合的“池”，使得不同池之间的结果变量存在差异（而同一池内无差异）。现有方法存在两种路径：(i) 在特定协变量关联假设下搜索单一的“最优”划分，或(ii) 从所有可能划分的全集中采样。这两种方法均忽略了一个现实：尤其在协变量存在相关结构时，大量可能的划分方式在统计上难以区分，却可能对政策或科学推断产生截然不同的影响。我们提出一种替代性视角——Rashomon划分集(RPS)。RPS中的每个元素采用树状几何结构对协变量空间进行划分。RPS包含所有后验值接近最大后验划分的划分方案（即便它们提供实质不同的解释），并采用不预设协变量间关联的先验分布——即ℓ₀先验，我们证明该先验具有极小极大最优性。基于RPS，我们可计算在属于RPS条件下，任何关于特征效应向量对结果的可测函数的后验分布。同时，我们刻画了相对于完整后验的近似误差界，并给出RPS大小的边界。模拟实验表明，该框架相较于传统正则化技术能产生更稳健的结论。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效应。