# Testing data binnings

## Abstract

Motivated by the question of data quantization and “binning,” we revisit the problem of identity testing of discrete probability distributions. Identity testing (a.k.a. one-sample testing), a fundamental and by now well-understood problem in distribution testing, asks, given a reference distribution (model) q and samples from an unknown distribution p, both over [n] = {1, 2,..., n}, whether p equals q, or is significantly different from it. In this paper, we introduce the related question of identity up to binning, where the reference distribution q is over k ≪ n elements: the question is then whether there exists a suitable binning of the domain [n] into k intervals such that, once “binned,” p is equal to q. We provide nearly tight upper and lower bounds on the sample complexity of this new question, showing both a quantitative and qualitative difference with the vanilla identity testing one, and answering an open question of Canonne [6]. Finally, we discuss several extensions and related research directions.