The Fisher metric on a manifold of probability distributions is usually treated as a metric on the tangent bundle. In this paper, we focus on the metric on the cotangent bundle induced from the Fisher metric with calling it the Fisher co-metric. We show that the Fisher co-metric can be defined directly without going through the Fisher metric by establishing a natural correspondence between cotangent vectors and random variables. This definition clarifies a close relation between the Fisher co-metric and the variance/covariance of random variables, whereby the Cram\'{e}r-Rao inequality is trivialized. We also discuss the monotonicity and the invariance of the Fisher co-metric with respect to Markov maps, and present a theorem characterizing the co-metric by the invariance, which can be regarded as a cotangent version of \v{C}encov's characterization theorem for the Fisher metric. The obtained theorem can also viewed as giving a characterization of the variance/covariance.

翻译：概率分布流形上的Fisher度量通常被视为切丛上的度量。本文聚焦于由Fisher度量诱导的余切丛上的度量，并将其称为Fisher余度量。我们通过建立余切向量与随机变量之间的自然对应关系，证明无需经由Fisher度量即可直接定义Fisher余度量。该定义厘清了Fisher余度量与随机变量方差/协方差之间的紧密联系，从而使得Cramér-Rao不等式变得平凡。我们还讨论了Fisher余度量关于马尔可夫映射的单调性与不变性，并提出了一个通过不变性刻画余度量的定理，该定理可视为Čencov关于Fisher度量特征化定理的余切版本。所得结果亦可视为对方差/协方差的一种特征化刻画。