Viterbi过程存在性的构造性证明
A constructive proof of the existence of Viterbi processes
摘要：
自数字通信的早期以来，隐马尔可夫模型（Hidden Markov Models，简称HMF）现在也被常规地用于语音识别、自然语言处理、图像处理和生物信息学。在隐马尔可夫模型$(X_i,Y_i)_{i\ge 1}$中，假设观测值$X_1,X_2,...$是条件独立的，给定一个“随机性”马尔可夫过程$Y_1,Y_2,...$，它本身是不可观测的;此外，$X_i$的条件分布仅取决于$Y_i$。HMM的理论和应用的核心是Viterbi算法，用于在给定观测数据$x_{1:n}$的情况下找到$Y_{1:n}$的{ \em最大后验}（MAP）估计$q_{1:n}=(q_1,q_2,...,q_n)$。最大{ \em后验}路径也称为维特比路径或比对。最近，已经尝试研究当$n\to \infty$。因此，它已被证明，在某些特殊情况下，一个明确的限制维特比对齐存在。虽然这些尝试是创新的，但它们依赖于相当强的假设，并涉及存在性的证据。这项工作证明了存在的无限维特比对齐在一个更建设性的方式和一个非常一般的类的障碍。
英文摘要：
Since the early days of digital communication, hidden Markov models (HMMs) have now been also routinely used in speech recognition, processing of natural languages, images, and in bioinformatics. In an HMM $(X_i,Y_i)_{i\ge 1}$, observations $X_1,X_2,...$ are assumed to be conditionally independent given an ``explanatory'' 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} (MAP) estimate $q_{1:n}=(q_1,q_2,...,q_n)$ of $Y_{1:n}$ given observed data $x_{1:n}$. Maximum {\em a posteriori} paths are also known as Viterbi paths or alignments. Recently, attempts have been made to study the behavior of Viterbi alignments when $n\to \infty$. Thus, it has been shown that in some special cases a well-defined limiting Viterbi alignment exists. While innovative, these attempts have relied on rather strong assumptions and involved proofs which are existential. This work proves the existence of infinite Viterbi alignments in a more constructive manner and for a very general class of HMMs.
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