Data fission: splitting a single data point

TL;DR

This paper introduces data fission, a new method inspired by Bayesian ideas, to split data into parts that together fully recover the original data, enabling tractable inference without sacrificing information.

Contribution

It proposes a general, finite-sample methodology for data splitting called data fission, extending previous approaches with a Bayesian-inspired continuous analog.

Findings

Data fission enables full data recovery from split parts in finite samples.

The method applies to post-selection inference in trend filtering and regression.

It offers a tractable alternative to traditional data splitting and p-value masking.

Abstract

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 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 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.