  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)
$\mathbf{q}$ and samples from an unknown distribution $\mathbf{p}$, both over
$[n]=\{1,2,\dots,n\}$, whether $\mathbf{p}$ equals $\mathbf{q}$, or is
significantly different from it.
  In this paper, we introduce the related question of 'identity up to binning,'
where the reference distribution $\mathbf{q}$ is over $k \ll n$ elements: the
question is then whether there exists a suitable binning of the domain $[n]$
into $k$ intervals such that, once "binned," $\mathbf{p}$ is equal to
$\mathbf{q}$. 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 (2019). Finally, we discuss several extensions and related
research directions.
