Analyticity, Convergence, and Convergence Rate of Recursive Maximum-Likelihood Estimation in Hidden Markov Models
Analyticity, Convergence, and Convergence Rate of Recursive Maximum-Likelihood Estimation in Hidden Markov Models
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DOI：
10.1109/tit.2010.2081110
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发表时间：
2009-04
影响因子：
2.5
通讯作者：
V. Tadic
中科院分区：
文献类型：
--
作者：
V. Tadic
本文考虑隐马尔可夫模型的递归最大似然估计量的渐近性质。本文重点研究渐进对数似然的解析性质以及递归最大似然估计器的点收敛和收敛速度。使用解析延拓原理，显示了解析参数化隐马尔可夫模型的渐近对数似然的解析性。依靠这一事实和微分几何（Lojasiewicz 不等式）的一些结果，证明了递归最大似然算法的几乎确定的点收敛，并导出了收敛速度的相对严格的界限。与隐马尔可夫模型中最大似然估计渐近行为的现有结果相反，本文的结果是在不假设对数似然函数具有孤立最大值（在该最大值处 Hessian 是严格负定的）的情况下获得的。
This paper considers the asymptotic properties of the recursive maximum-likelihood estimator for hidden Markov models. The paper is focused on the analytic properties of the asymptotic log-likelihood and on the point-convergence and convergence rate of the recursive maximum-likelihood estimator. Using the principle of analytic continuation, the analyticity of the asymptotic log-likelihood is shown for analytically parameterized hidden Markov models. Relying on this fact and some results from differential geometry (Lojasiewicz inequality), the almost sure point convergence of the recursive maximum-likelihood algorithm is demonstrated, and relatively tight bounds on the convergence rate are derived. As opposed to the existing result on the asymptotic behavior of maximum-likelihood estimation in hidden Markov models, the results of this paper are obtained without assuming that the log-likelihood function has an isolated maximum at which the Hessian is strictly negative definite.