Suppose we observe a random vector $X$ from some distribution $P$ in a known family with unknown parameters. We ask the following question: when is it possible to split $X$ into two parts $f(X)$ and $g(X)$ such that neither part is sufficient to reconstruct $X$ by itself, but both together can recover $X$ fully, and the joint distribution of $(f(X),g(X))$ is tractable? As one example, if $X=(X_1,\dots,X_n)$ and $P$ is a product distribution, then for any $m<n$, we can split the sample to define $f(X)=(X_1,\dots,X_m)$ and $g(X)=(X_{m+1},\dots,X_n)$. Rasines and Young (2022) offers an alternative route of accomplishing this task through randomization of $X$ with additive Gaussian noise which enables post-selection inference in finite samples for Gaussian distributed data and asymptotically for non-Gaussian additive models. In this paper, 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 p-value masking. We exemplify the method on a few prototypical applications, such as post-selection inference for trend filtering and other regression problems.

翻译：假设我们观测到一个随机向量$X$，其来自某个已知分布族$P$但参数未知。我们提出以下问题：何时能够将$X$分割为两部分$f(X)$和$g(X)$，使得任一部分本身均不足以重建$X$，但两者结合可完全恢复$X$，且$(f(X),g(X))$的联合分布易于处理？例如，若$X=(X_1,\dots,X_n)$且$P$为乘积分布，则对任意$m<n$，可对样本进行分割，定义$f(X)=(X_1,\dots,X_m)$和$g(X)=(X_{m+1},\dots,X_n)$。Rasines与Young（2022）提出另一种实现路径：通过加性高斯噪声对$X$进行随机化，从而在高斯分布数据中实现有限样本下的选择后推断，并在非高斯加性模型中渐近成立。本文通过借鉴贝叶斯推断的思想，提出一种更通用的有限样本分割方法，该方法可视为数据分割的连续类比，并给出频率学派解。我们将此方法称为"数据裂变"，以区别于数据分割、数据雕刻与p值掩蔽。最后，我们通过趋势过滤及其他回归问题中的选择后推断等典型应用示例对该方法进行说明。