To characterize the Kullback-Leibler divergence and Fisher information in general parametrized hidden Markov models, in this paper, we first show that the log likelihood and its derivatives can be represented as an additive functional of a Markovian iterated function system, and then provide explicit characterizations of these two quantities through this representation. Moreover, we show that Kullback-Leibler divergence can be locally approximated by a quadratic function determined by the Fisher information. Results relating to the Cram\'{e}r-Rao lower bound and the H\'{a}jek-Le Cam local asymptotic minimax theorem are also given. As an application of our results, we provide a theoretical justification of using Akaike information criterion (AIC) model selection in general hidden Markov models. Last, we study three concrete models: a Gaussian vector autoregressive-moving average model of order $(p,q)$, recurrent neural networks, and temporal restricted Boltzmann machine, to illustrate our theory.

翻译：为刻画一般参数化隐马尔可夫模型中的库尔贝克-莱布勒散度与Fisher信息量，本文首先证明对数似然及其导数可表示为马尔可夫迭代函数系统的加性泛函，进而通过该表示给出两者的显式刻画。此外，我们证明库尔贝克-莱布勒散度可由Fisher信息量决定的二次函数局部逼近。文中还给出了与Cramér-Rao下界及Hájek-Le Cam局部渐近极小极大定理相关的结果。作为理论应用，我们为广义隐马尔可夫模型中采用赤池信息准则（AIC）进行模型选择提供了理论依据。最后，通过三类具体模型——高斯向量自回归滑动平均模型（阶数(p,q)）、循环神经网络及时序受限玻尔兹曼机——验证理论的有效性。