论文标题
无似然假设检验
Likelihood-free hypothesis testing
论文作者
论文摘要
考虑二元假设检验的问题。给定的$ z $来自$ \ mathbb p^{\ otimes m} $或$ \ mathbb q^{\ otimes m} $，以在两者之间决定有较小的错误可能性，而在许多情况下，在许多情况下，必须具有$ m \ m \ m \ asymp1/ε^2 $ $ $ $ $ $ quans $ mathion $ quyb，$ mather $ （$ \ mathsf {tv} $）。但是，实现这一目标需要完全了解分布，例如使用Neyman-Pearson测试。在本文中，我们考虑了我们称之为无可能的假设测试的问题的变化，其中访问$ \ Mathbb P $和$ \ Mathbb Q $是通过$ n $ i.i.d提供的。每个观察结果。如果假定$ \ mathbb p $和$ \ mathbb q $属于非参数家族，我们证明存在$ n $ n $和$ m $之间的基本权衡，由$ nm \ asymp n_ asymp n_ asmp n_ \ sf \ sf \ sf {gof}^2（ε）$ n _ i i ips $ testing testing smin testing sime testime smins $ sim pysim sim pysim sim sim sim sim pessing（gof}（ε假设$ h_0：\，\ mathbb p = \ mathbb q $ vs $ h_1：\，\ mathsf {tv}（\ mathbb p，\ mathbb q）\geqε$。除了所有离散分布的家族外，我们还为三个分布族展示了这一点，我们为此获得了更为复杂的权衡，展现出额外的相转换。我们的结果表明，在不完全估计$ \ mathbb p $和$ \ mathbb Q $的情况下进行测试的可能性，提供了$ m \ gg 1/ε^2 $。
Consider the problem of binary hypothesis testing. Given $Z$ coming from either $\mathbb P^{\otimes m}$ or $\mathbb Q^{\otimes m}$, to decide between the two with small probability of error it is sufficient, and in many cases necessary, to have $m\asymp1/ε^2$, where $ε$ measures the separation between $\mathbb P$ and $\mathbb Q$ in total variation ($\mathsf{TV}$). Achieving this, however, requires complete knowledge of the distributions and can be done, for example, using the Neyman-Pearson test. In this paper we consider a variation of the problem which we call likelihood-free hypothesis testing, where access to $\mathbb P$ and $\mathbb Q$ is given through $n$ i.i.d. observations from each. In the case when $\mathbb P$ and $\mathbb Q$ are assumed to belong to a non-parametric family, we demonstrate the existence of a fundamental trade-off between $n$ and $m$ given by $nm\asymp n_\sf{GoF}^2(ε)$, where $n_\sf{GoF}(ε)$ is the minimax sample complexity of testing between the hypotheses $H_0:\, \mathbb P=\mathbb Q$ vs $H_1:\, \mathsf{TV}(\mathbb P,\mathbb Q)\geqε$. We show this for three families of distributions, in addition to the family of all discrete distributions for which we obtain a more complicated trade-off exhibiting an additional phase-transition. Our results demonstrate the possibility of testing without fully estimating $\mathbb P$ and $\mathbb Q$, provided $m \gg 1/ε^2$.
