Data fission: splitting a single data point

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 = (X1, . . . , Xn) and P is a product distribution, then for any m < n, we can split the sample to define f (X) = (X1, . . . , Xm) and g(X) = (Xm+1, . . . , Xn). Rasines and Young (2021) 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 blurring, 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.

Generalized Data Thinning Using Sufficient Statistics

“…Then, U ∼ N n (θ, (1+γ)I n ) and V ∼ N n (θ, (1+γ −1 )I n ) are independent. Rasines & Young (2022) and Leiner et al (2022) applied this decomposition to address Scenario 1 in Section 1. Additionally, Rasines & Young (2022) showed that this leads to asymptotically valid inference under certain regularity conditions, even when X is not normally distributed.…”

“…Our generalized thinning framework achieves the goal set out in their paper, providing a concrete recipe for finding such decompositions in a broad set of examples. Another paper with a similar goal is Leiner et al (2022). They define "data fission", which seeks to find random variables f (X) and g(X) for which the distributions of f (X) and g(X) | f (X) are known and for which X = h(f (X), g(X)).…”