  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.
