In this paper, we propose a novel equivalence between probability theory and information theory. For a single random variable, Shannon's self-information, $I=-\log {p}$, is an alternative expression of a probability $p$. However, for two random variables, no information equivalent to the $p$-value has been identified. Here, we prove theorems that demonstrate that mutual information (MI) is equivalent to the $p$-value irrespective of prior information about the distribution of the variables. If the maximum entropy principle can be applied, our equivalence theorems allow us to readily compute the $p$-value from multidimensional MI. By contrast, in a contingency table of any size with known marginal frequencies, our theorem states that MI asymptotically coincides with the logarithm of the $p$-value of Fisher's exact test, divided by the sample size. Accordingly, the theorems enable us to perform a meta-analysis to accurately estimate MI with a low $p$-value, thereby calculating informational interdependence that is robust against sample size variation. Thus, our theorems demonstrate the equivalence of the $p$-value and MI at every dimension, use the merits of both, and provide fundamental information for integrating probability theory and information theory.

翻译：本文提出了概率论与信息论之间的一种新颖等价关系。对于单一随机变量，香农自信息量 $I=-\log {p}$ 是概率 $p$ 的另一种表达形式。然而，对于两个随机变量，尚未发现与 $p$ 值等价的信息量。本文证明了若干定理，表明无论变量分布的先验信息如何，互信息量均与 $p$ 值等价。若最大熵原理适用，该等价性定理可便捷地通过多维互信息量计算 $p$ 值。相反，在任意大小且边际频数已知的列联表中，本定理指出互信息量渐近等于费希尔精确检验的 $p$ 值对数除以样本容量。据此，定理可实现对低 $p$ 值情形下互信息量的精确估计，从而进行元分析，计算出对样本容量变化具有稳健性的信息相互依存度。因此，本文定理证明了 $p$ 值与互信息量在各个维度上的等价性，充分利用了两者的优势，并为整合概率论与信息论提供了基础性信息。