发散预测状态：平稳、遍历隐马尔可夫过程的统计复杂性维数
Divergent Predictive States: The Statistical Complexity Dimension of Stationary, Ergodic Hidden Markov Processes
摘要：
即使是简单定义的有限状态生成器也会产生随机过程，这些随机过程需要跟踪不可数的无穷大概率特征以进行最佳预测。对于隐马尔可夫链产生的过程，后果是戏剧性的。他们的预测模型一般是无限状态的。而且，直到最近，人们才能确定它们的内在随机性和结构复杂性。然而，前传引入了精确计算香农熵率（随机性）的方法，并建设性地确定了它们的最小（尽管是无限的）预测特征集。利用这一点，我们解决了互补的挑战，确定如何结构化的隐马尔可夫过程是通过计算其统计复杂性维度-信息维度的最小集的预测功能。它跟踪最佳预测一大类真正复杂的进程所需的最小内存资源的发散率。
英文摘要：
Even simply-defined, finite-state generators produce stochastic processes that require tracking an uncountable infinity of probabilistic features for optimal prediction. For processes generated by hidden Markov chains the consequences are dramatic. Their predictive models are generically infinite-state. And, until recently, one could determine neither their intrinsic randomness nor structural complexity. The prequel, though, introduced methods to accurately calculate the Shannon entropy rate (randomness) and to constructively determine their minimal (though, infinite) set of predictive features. Leveraging this, we address the complementary challenge of determining how structured hidden Markov processes are by calculating their statistical complexity dimension -- the information dimension of the minimal set of predictive features. This tracks the divergence rate of the minimal memory resources required to optimally predict a broad class of truly complex processes.
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