“Data Fission: Splitting a Single Data Point” by Leiner et al.

(James Leiner and Ramdas 2023). The arXiv version is actually a bit more comfortable to read. Abstract:

Suppose we observe a random vector \(X\) from some distribution in a known family with unknown parameters. We ask the following question: when is it possible to split \(X\) into two pieces \(f(X)\) and \(g(X)\) 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 \(p\)-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.

The paper offers a clear review and systematization of older work, most prominently (Rasines and Young 2022) (cited in the abstract), with some useful generalizations.

The idea is cool, but I find the applications to practical regression cases given in the paper somewhat… impractical. For usual linear regression with a continuous response, the applicability of the method relies on (1) noise being homoskedastic and gaussian, (2) the existence of a consistent estimator \(\hat \sigma\) of noise variance, and (3) samples being large enough (guarantees are only asymptotic). On the other hand, in the theoretically simpler case of logistic regression, there’s a technical complication in that, under the usual GLM assumption \(\theta(X) = X\beta\), the relevant log-likelihood for maximization in the inferential stage is not a concave function of \(\beta\), possibly hindering optimization. If I got it right, the authors suggest to ignore the conditional dependence of \(g(Y_i)\) on \(f(Y_i)\) to circumvent these complications (see Appendix E.4), which I honestly don’t understand.

A case in which planets align and results have a nice analytic form is that of Poisson regression, for which I will sketch the idea in some detail. Suppose that we are given data \(\mathcal D _0 = \{(X_i,Y_i)\}_{i=1}^N\) independently drawn from a joint \((X,Y)\) distribution, and we assume \(Y \vert X \sim \text{Pois}(\lambda (X))\) for some unknown function \(\lambda (X)\) we would like to model. The key observation is (cf. Appendix A of the reference) that if \(Z \vert Y \sim \text{Binom}(Y,\,p)\), then \(Z \sim \text {Pois}(p\lambda)\) and \(\overline Z = Y - Z \sim \text{Pois}((1-p)\lambda)\), with \(Z\) and \(\overline Z\) unconditionally independent. Hence, if we randomly draw \(Z _i\) according to \(\text{Binom}(Y_i,\,p)\), and set \(\overline Z _i = Y_i -Z_i\), the two datasets \(\mathcal D = \{(X_i,\,Z_i)\}\) and \(\overline{\mathcal D} = \{(X_i,\,\overline Z_i)\}\), are conditionally independent given the observed covariates \(X_i\). This allows to decouple different aspects of modeling, such as model selection and inference, avoiding the usual biases associated with the intrinsic randomness of the selection step.

The authors focus on regression with fixed covariates, because in that setting the simpler option of data-splitting is less motivated, calling for alternatives. However, the method can be applied equally well to deal with selective inference in random covariates settings, since it leads - at least in principle - to inferences which are valid conditionally on the observed covariates and (in the general case) the randomized responses \(f(Y_i)\) of the selection stage.

References

James Leiner, Larry Wasserman, Boyan Duan, and Aaditya Ramdas. 2023. “Data Fission: Splitting a Single Data Point.” Journal of the American Statistical Association 0 (0): 1–12. https://doi.org/10.1080/01621459.2023.2270748.

Rasines, D García, and G A Young. 2022. “Splitting strategies for post-selection inference.” Biometrika 110 (3): 597–614. https://doi.org/10.1093/biomet/asac070.