Statistical inference from data is foundational task in science. Recently, it receives growing attention for its central role in inference systems of primary interest in data science, artificial intelligence, or machine learning. However, the understanding of statistical inference itself is not that solid while regarded as a matter of subjective choice or implemented in obscure ways. We here show that statistical inference has rigorous scientific description for long sequence of exchangeable binary random variables, the prototypal stochasticity in theories and applications. A linear differential equation is derived from the exchangeability, and it turns out that statistical inference is given by the Green's functions. Our finding is the answer to the normative and foundational issue in science, and its significance will be far-reaching in all pure and applied fields.

翻译：从数据得出的统计推论是科学的基础任务。最近,它因其在数据科学、人工智能或机器学习方面主要感兴趣的推论体系中的核心作用而日益受到重视。然而,对统计推论本身的理解并不是那么牢固,而是被视为主观选择的问题或以模糊的方式加以执行。我们在这里表明,统计推论具有严格的科学描述,可以交换的二进随机变量的长序列,即理论和应用中的原生性随机变量。线性差异方程式来自可交换性,它证明统计推论是由绿方的功能提供的。我们的发现是对科学中的规范和基础问题的答案,其意义将在所有纯科学和应用领域产生深远的影响。