Universal Hypothesis Testing with Kernels: Asymptotically Optimal Tests for Goodness of Fit
Universal Hypothesis Testing with Kernels: Asymptotically Optimal Tests for Goodness of Fit
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发表时间：
2018-02
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通讯作者：
Shengyu Zhu;Biao Chen;Pengfei Yang;Zhitang Chen
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作者：
Shengyu Zhu;Biao Chen;Pengfei Yang;Zhitang Chen
我们描述了非参数拟合优度检验的渐近性能。 II 类错误概率的指数衰减率用作渐近性能度量，并且如果在 I 类错误概率的恒定水平约束下达到最大速率，则测试是最佳的。我们表明，两类基于最大平均差异（MMD）的测试在 $\mathbb R^d$ 上达到了这种最优性，而基于二次时间核斯坦因差异（KSD）的测试在宽松的水平约束下实现了最大指数衰减率。在相同的性能指标下，我们继续证明，只要核是有界连续且具有特征的，基于二次时间 MMD 的双样本检验对于一般双样本问题也是最优的。我们方法的关键是来自大偏差理论的萨诺夫定理以及 MMD 和 KSD 的弱可度量特性。
We characterize the asymptotic performance of nonparametric goodness of fit testing. The exponential decay rate of the type-II error probability is used as the asymptotic performance metric, and a test is optimal if it achieves the maximum rate subject to a constant level constraint on the type-I error probability. We show that two classes of Maximum Mean Discrepancy (MMD) based tests attain this optimality on $\mathbb R^d$, while the quadratic-time Kernel Stein Discrepancy (KSD) based tests achieve the maximum exponential decay rate under a relaxed level constraint. Under the same performance metric, we proceed to show that the quadratic-time MMD based two-sample tests are also optimal for general two-sample problems, provided that kernels are bounded continuous and characteristic. Key to our approach are Sanov's theorem from large deviation theory and the weak metrizable properties of the MMD and KSD.