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.