We study the problem of parameter estimation in time series stemming from general stochastic processes, where the outcomes may exhibit arbitrary temporal correlations. In particular, we address the question of how much Fisher information is lost if the stochastic process is compressed into a single histogram, known as the empirical distribution. As we show, the answer is non-trivial due to the correlations between outcomes. We derive practical formulas for the resulting Fisher information for various scenarios, from generic stationary processes to discrete-time Markov chains to continuous-time classical master equations. The results are illustrated with several examples.

翻译：我们研究源自一般随机过程的时间序列中的参数估计问题，其中结果可能表现出任意的时间相关性。特别地，我们探讨了当随机过程被压缩为单一直方图（即经验分布）时，会损失多少费舍尔信息。正如我们所展示的，由于结果之间的相关性，答案并非显而易见。我们推导了针对各种场景下所得费舍尔信息的实用公式，涵盖从一般平稳过程到离散时间马尔可夫链，再到连续时间经典主方程。通过多个示例对结果进行了说明。