论文标题
切尔诺夫型相对熵的经验概率浓度
Chernoff-type Concentration of Empirical Probabilities in Relative Entropy
论文作者
论文摘要
我们研究了经验概率矢量相对于$ K $类别的多项式采样中的真实概率向量的相对熵，当$ K $类别的多项式采样中，当将样本量$ n $乘以时，这也是对数可能的比例统计量。我们概括了最近的结果，并表明该统计量的力矩生成函数在单位间隔上由$ n $的多项式界定，而不是所有真实概率向量。我们描述了由$（k，n）$索引的多项式族的家族并获得明确的公式。因此，我们开发了Chernoff型尾巴边界，包括来自界限最小化器的大型样品扩展的封闭形式。我们的界限主导了经典的类型方法，并且与最先进的状态具有竞争力。我们证明了估计看不见的蝴蝶比例的应用。
We study the relative entropy of the empirical probability vector with respect to the true probability vector in multinomial sampling of $k$ categories, which, when multiplied by sample size $n$, is also the log-likelihood ratio statistic. We generalize a recent result and show that the moment generating function of the statistic is bounded by a polynomial of degree $n$ on the unit interval, uniformly over all true probability vectors. We characterize the family of polynomials indexed by $(k,n)$ and obtain explicit formulae. Consequently, we develop Chernoff-type tail bounds, including a closed-form version from a large sample expansion of the bound minimizer. Our bound dominates the classic method-of-types bound and is competitive with the state of the art. We demonstrate with an application to estimating the proportion of unseen butterflies.
