标记马尔可夫链的总变异距离

Taolue Chen, S. Kiefer
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引用次数: 31

摘要

标记马尔可夫链(lmc)广泛应用于概率验证、语音识别、计算生物学等领域。检验两个lmc的等价性是一个经典问题,需要进行广泛的研究,而总变异距离为两个lmc的“不等价性”提供了一个自然的度量:它是lmc分配给同一事件的概率之间的最大差异。在本文中,我们发展了两个lmc之间总变异距离的理论,重点是算法方面:(1)我们提供了一个多项式时间算法来确定两个lmc是否具有距离1,即它们是否几乎总是可以被区分;(2)给出了一种任意精度的距离逼近算法;(3)我们证明了阈值问题,即距离是否超过给定的阈值,对于平方根和问题来说是np困难的。我们还在总变异距离和伯努利卷积之间建立了联系。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On the total variation distance of labelled Markov chains
Labelled Markov chains (LMCs) are widely used in probabilistic verification, speech recognition, computational biology, and many other fields. Checking two LMCs for equivalence is a classical problem subject to extensive studies, while the total variation distance provides a natural measure for the "inequivalence" of two LMCs: it is the maximum difference between probabilities that the LMCs assign to the same event. In this paper we develop a theory of the total variation distance between two LMCs, with emphasis on the algorithmic aspects: (1) we provide a polynomial-time algorithm for determining whether two LMCs have distance 1, i.e., whether they can almost always be distinguished; (2) we provide an algorithm for approximating the distance with arbitrary precision; and (3) we show that the threshold problem, i.e., whether the distance exceeds a given threshold, is NP-hard and hard for the square-root-sum problem. We also make a connection between the total variation distance and Bernoulli convolutions.
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