Infinite Viterbi alignments in the two state hidden Markov models
J. Lember, A. Koloydenko
备注：
Several minor changes and corrections have been made in the arguments as suggested by anonymous reviewers, which should hopefully improve readability. Abstract has been added
发表信息：
Acta et Commentationes Universitatis Tartuensis de Mathematica, Volume 12, 2008, pp. 109-124
Since the early days of digital communication, Hidden Markov Models (HMMs) have now been routinely used in speech recognition, processing of natural languages, images, and in bioinformatics. An HMM $(X_i,Y_i)_{i\ge 1}$ assumes observations $X_1,X_2,...$ to be conditionally independent given an "explanotary" Markov process $Y_1,Y_2,...$, which itself is not observed; moreover, the conditional distribution of $X_i$ depends solely on $Y_i$. Central to the theory and applications of HMM is the Viterbi algorithm to find {\em a maximum a posteriori} estimate $q_{1:n}=(q_1,q_2,...,q_n)$ of $Y_{1:n}$ given the observed data $x_{1:n}$. Maximum {\em a posteriori} paths are also called Viterbi paths or alignments. Recently, attempts have been made to study the behavior of Viterbi alignments of HMMs with two hidden states when $n$ tends to infinity. It has indeed been shown that in some special cases a well-defined limiting Viterbi alignment exists. While innovative, these attempts have relied on rather strong assumptions. This work proves the existence of infinite Viterbi alignments for virtually any HMM with two hidden states.
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我们解决了可变长度隐马尔可夫模型中的上下文树估计问题。我们提出了一个估计的上下文树的隐马尔可夫过程，它不需要事先上界的深度上下文树。我们证明了估计量是强相合的。这一方法使用了Finesso和Lorenzo（Consistent estimation of the order for Markov and hidden Markov chains（1990））和E.Gassiat和S.Boucheron（Optimal error exponents in hidden Markov model order estimation（2003））的信息论混合不等式。我们提出了一种算法来有效地计算估计，并提供模拟研究，以支持我们的结果。
