Generalized Data Thinning Using Sufficient Statistics

Abstract

Our goal is to develop a general strategy to decompose a random variable X into multiple independent random variables, without sacrificing any information about unknown parameters. A recent paper showed that for some well-known natural exponential families, X can be thinned into independent random variables $X^{(1)},\dots ,X^{(K)}$, such that $X={\sum }_{k=1}^K{X}^{(k)}$. These independent random variables can then be used for various model validation and inference tasks, including in contexts where traditional sample splitting fails. In this article, we generalize their procedure by relaxing this summation requirement and simply asking that some known function of the independent random variables exactly reconstruct X. This generalization of the procedure serves two purposes. First, it greatly expands the families of distributions for which thinning can be performed. Second, it unifies sample splitting and data thinning, which on the surface seem to be very different, as applications of the same principle. This shared principle is sufficiency. We use this insight to perform generalized thinning operations for a diverse set of families. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Keywords:

• Cross-validation
• Exponential families
• Model validation
• Sample splitting
• Selective inference

Disclosure Statement

The authors have no relevant financial or nonfinancial competing interests to report.

Acknowledgments

We thank Nicholas Irons for identifying a problem with a previous version of the proof about Bernoulli thinning (Section 6.1).

Notes

1 Although θ is often used in the gamma distribution to denote the scale parameter, here we use it to denote the rate parameter.

Additional information

Funding

The authors gratefully acknowledge funding from NIH R01 EB026908, NIH R01 DA047869, NIH 5P30DA048736, ONR N00014-23-1-2589, NSF DMS 2322920, a Simons Investigator Award in Mathematical Modeling of Living Systems, and the Keck Foundation to DW; NIH R01 GM123993 to DW and JB; a Natural Sciences and Engineering Research Council of Canada Discovery Grant to LG; and a Natural Sciences and Engineering Research Council of Canada Postgraduate Scholarship-Doctoral to AD.