标题： 隐马尔可夫过程：一种新的表示、熵率和估计熵 显示英文标题

标题： Hidden Markov Process: A New Representation, Entropy Rate and Estimation Entropy

摘要： 我们考虑一对相关过程{Z_n}和{S_n}（双侧），其中前者是可观测的，后者是隐藏的。 在根据其有限过去历史估计 Z_n 时存在不确定性，即 H(Z_n|Z_0^{n-1})，而在根据此观测估计 S_n 时存在不确定性，即 H(S_n|Z_0^{n-1})，它们都是 n 的序列。这些序列的极限（以及其存在性）具有实际和理论上的兴趣。 如果存在第一个极限，则为熵率。 我们将第二个极限称为估计熵。 一个与另一个过程联合相关的例子是隐马尔可夫过程。 它是马尔可夫状态过程的无记忆观测，其中状态转移独立于过去的观测。 我们考虑使用迭代函数系统对隐马尔可夫过程的新表示。 在此表示中，状态转移与过程确定性相关。 这种表示为该过程的两个极限熵的分析提供了统一的框架，从而得到了极限的积分表达式。 此分析表明，在较弱的条件下，极限存在，并提供了一种计算相应序列元素的简单方法。 显示英文摘要

摘要： We consider a pair of correlated processes {Z_n} and {S_n} (two sided), where the former is observable and the later is hidden. The uncertainty in the estimation of Z_n upon its finite past history is H(Z_n|Z_0^{n-1}), and for estimation of S_n upon this observation is H(S_n|Z_0^{n-1}), which are both sequences of n. The limits of these sequences (and their existence) are of practical and theoretical interest. The first limit, if exists, is the entropy rate. We call the second limit the estimation entropy. An example of a process jointly correlated to another one is the hidden Markov process. It is the memoryless observation of the Markov state process where state transitions are independent of past observations. We consider a new representation of hidden Markov process using iterated function system. In this representation the state transitions are deterministically related to the process. This representation provides a unified framework for the analysis of the two limiting entropies for this process, resulting in integral expressions for the limits. This analysis shows that under mild conditions the limits exist and provides a simple method for calculating the elements of the corresponding sequences.