有限长个体测试序列的通用分类器的最优性
On the optimality of universal classifiers for finite-length individual test sequences
摘要：
我们认为对有限长度的个人序列，是实现未知的，有限的字母表，在一类M的源与消失的记忆（例如平稳马尔可夫源）的平稳源。通用分类器的任务是决定两个序列是从相同的源出现还是从M中的两个不同的源出现，并且它必须在没有两个潜在概率度量的任何先验知识的情况下执行此任务。给定一个保真度函数和一个保真度准则，定义了一个通用分类器的分类错误概率。针对$N$ -序列对定义了两个通用分类器：“经典”固定长度（FL）通用分类器和替代可变长度（VL）通用分类器。根据Wyner和Ziv（1996），证明了如果单个序列的长度N小于由类M的属性确定的截止值，则任何通用分类器都将以高概率失败。它表明，N的值大于截止率，相对于两个分类器中的任何一个的分类误差趋于零的序列的长度趋于无穷大。然而，对于任何有限长度，与可变长度通用分类器相关联的分类错误的概率均匀地小于（或等于）与“经典”固定长度通用分类器相关联的分类错误的概率。
英文摘要：
We consider pairs of finite-length individual sequences that are realizations of unknown, finite alphabet, stationary sources in a clas M of sources with vanishing memory (e.g. stationary Markov sources). The task of a universal classifier is to decide whether the two sequences are emerging from the same source or are emerging from two distinct sources in M, and it has to carry this task without any prior knowledge of the two underlying probability measures. Given a fidelity function and a fidelity criterion, the probability of classification error for a given universal classifier is defined. Two universal classifiers are defined for pairs of $N$ -sequence: A "classical" fixed-length (FL) universal classifier and an alternative variable-length (VL) universal classifier. Following Wyner and Ziv (1996) it is demonstrated that if the length of the individual sequences N is smaller than a cut-off value that is determined by the properties of the class M, any universal classifier will fail with high probability . It is demonstrated that for values of N larger than the cut-off rate, the classification error relative to either one of the two classifiers tends to zero as the length of the sequences tends to infinity. However, the probability of classification error that is associated with the variable-length universal classifier is uniformly smaller (or equal) to the one that is associated with the "classical" fixed-length universal classifier, for any finite length.
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