We propose a generalised Fisher information or a one-parameter extended class of the Fisher information for the case of one random variable. This new form of the Fisher information is obtained from the intriguing connection between the standard Fisher information and the variational principle together with the nonuniqueness property of the Lagrangian. A generalised Cramér--Rao inequality is also derived and a Fisher information hierarchy is also obtained from the two-parameter Kullback-Leibler divergence. An interesting point is that the whole Fisher information hierarchy, except for the standard Fisher information, does not follow the additive rule. Furthermore, the idea can be directly extended to obtain the one-parameter generalised Fisher information matrix for the case of one random variable with multi-estimated parameters. The hierarchy of the Fisher information matrices is obtained. The geometrical meaning of the first two matrices in the hierarchy is studied through the normal distribution. What we find is that these first two Fisher matrices give different nature of curvature on the same statistical manifold for the normal distribution.

对于一个随机变量的情况，我们提出了广义 Fisher 信息或 Fisher 信息的单参数扩展类。这种新形式的 Fisher 信息是从标准 Fisher 信息和变分原理之间的有趣联系以及拉格朗日量的非唯一性属性中获得的。还推导了广义 Cramér-Rao 不等式，并且还从双参数 Kullback-Leibler 散度获得了 Fisher 信息层次结构。有趣的一点是，整个Fisher信息层次结构，除了标准的Fisher信息外，并不遵循加法规则。此外，该思想可以直接扩展，得到单参数广义Fisher信息矩阵，用于单随机变量多估计参数的情况。获得 Fisher 信息矩阵的层次结构。通过正态分布研究层次中前两个矩阵的几何意义。我们发现，前两个 Fisher 矩阵在正态分布的同一统计流形上给出了不同的曲率性质。