测试数据分组

摘要：

出于数据量化和“装箱”的问题，我们重新审视离散概率分布的身份测试的问题。鉴别试验（又名单样本检验）是分布检验中的一个基本的并且到目前为止已经被很好理解的问题，它询问给定参考分布（模型）$\mathbf{q}$和来自未知分布$\mathbf{p}$的样本，两者都在$[n]=\{1,2,\dots,n\}$上，$\mathbf{p}$是否等于$\mathbf{q}$，或者与之显著不同。本文介绍了相关的"直到装箱的同一性“问题，其中参考分布$\mathbf{q}$在$k \ll n$元素上：问题是是否存在域$[n]$到$k$间隔的合适的装仓，使得一旦“装仓”，$\mathbf{p}$等于$\mathbf{q}$。我们提供了这个新问题的样本复杂性的几乎严格的上限和下限，显示了与香草身份测试的定量和定性差异，并回答了Canonne（2019）的一个开放问题。最后，我们讨论了几个扩展和相关的研究方向。

英文摘要：

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.

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