Data Fission: Splitting a Single Data Point

Suppose we observe a random vector <i>X</i> from some distribution in a known family with unknown parameters. We ask the following question: when is it possible to split <i>X</i> into two pieces <i>f</i>(<i>X</i>) and <i>g</i>(<i>X</i>) such that neither part is sufficient to reconstruct X by itself, but both together can recover X fully, and their joint distribution is tractable? One common solution to this problem when multiple samples of X are observed is data splitting, but Rasines and Young offers an alternative approach that uses additive Gaussian noise—this enables post-selection inference in finite samples for Gaussian distributed data and asymptotically when errors are non-Gaussian. In this article, we offer a more general methodology for achieving such a split in finite samples by borrowing ideas from Bayesian inference to yield a (frequentist) solution that can be viewed as a continuous analog of data splitting. We call our method data fission, as an alternative to data splitting, data carving and <i>p</i>-value masking. We exemplify the method on several prototypical applications, such as post-selection inference for trend filtering and other regression problems, and effect size estimation after interactive multiple testing. Supplementary materials for this article are available online.

假设我们从某个含未知参数的已知分布族中观测到一个随机向量X，提出如下问题：能否将X拆分为f(X)与g(X)两部分，使得任一单一部分均无法单独重构X，但二者结合可完全还原X，且二者的联合分布易于处理？当可获取X的多个样本时，解决该问题的常见方案是数据拆分（data splitting），但拉西内斯（Rasines）与杨（Young）提出了一种使用加性高斯噪声的替代方法——该方法可在有限样本下针对高斯分布数据开展选择后推断，且在误差非高斯时也能实现渐近推断。本文提出一种更具普适性的方法，可在有限样本下实现此类拆分：我们借鉴贝叶斯推断（Bayesian inference）的思想，得到了可视为数据拆分连续类比形式的（频率学派）解法。我们将该方法命名为数据裂变（data fission），作为数据拆分、数据雕琢（data carving）与p值掩码（p-value masking）的替代方案。我们通过多个典型应用场景对该方法进行演示，例如趋势滤波与其他回归问题的选择后推断，以及交互式多重检验后的效应量估计。本文的补充材料可在线获取。