Abstract

In this article, we study sequential testing problems with overlapping hypotheses. We first focus on the simple problem of assessing if the mean μ of a Gaussian distribution is smaller or larger than a fixed $ϵ>0;$ if $\mu \in (-ϵ,ϵ),$ both answers are considered to be correct. Then, we consider probably approximately correct best arm identification in a bandit model: given K probability distributions on $\mathbb{R}$ with means ${\mu }_1,\dots ,{\mu }_K,$ we derive the asymptotic complexity of identifying, with risk at most δ, an index $I\in \{1,\dots ,K\}$ such that ${\mu }_I\ge {\max }_i{\mu }_i-ϵ.$ We provide nonasymptotic bounds on the error of a parallel general likelihood ratio test, which can also be used for more general testing problems. We further propose a lower bound on the number of observations needed to identify a correct hypothesis. Those lower bounds rely on information-theoretic arguments, and specifically on two versions of a change of measure lemma (a high-level form and a low-level form) whose relative merits are discussed.

摘要

在本文中，我们研究具有重叠假设的顺序测试问题。我们首先关注评估高斯分布的均值μ小于或大于固定点的简单问题$ϵ>0;$ 如果 $\mu \in （-ϵ，ϵ），$这两个答案都被认为是正确的。然后，我们考虑在强盗模型中可能近似正确的最佳手臂识别：给定K个概率分布$\mathbb{[R}$ 用手段 ${\mu }_{\mathrm{1个}}，\dots ，{\mu }_{ķ}，$我们得出了风险最大为δ的指标的识别的渐近复杂度$\mathrm{一世}\in \{\mathrm{1个}，\dots ，ķ\}$ 这样 ${\mu }_{\mathrm{一世}}\ge {\mathrm{最大限度}}_{\mathrm{一世}}{\mu }_{\mathrm{一世}}-ϵ。$我们提供了关于并行一般似然比检验的误差的非渐近边界，它也可以用于更一般的检验问题。我们进一步提出了确定正确假设所需的观测值数量的下限。这些下限依赖于信息理论论证，特别是依赖于衡量度量引理变化的两个版本（高级形式和低级形式），它们的相对优点在此进行了讨论。