统计推断为格林函数
Statistical inference as Green's functions
摘要：
从数据中进行统计推断是科学中的一项基本任务。最近，它因其在数据科学和机器学习中主要感兴趣的推理系统中的核心作用而受到越来越多的关注。然而，对统计推断的理解并不那么牢固，仍然是主观信念的问题，或者是曾经声称客观的常规程序。本文证明了可交换二元随机变量长序列的统计推断、原型随机性在理论和应用上都有一个客观的描述。一个线性微分方程是从被称为德菲内蒂表示定理的恒等式推导出来的，结果证明统计推断是由格林函数给出的。我们的发现是对科学以数据为基础追求客观性的规范性问题的回答，其意义在大多数纯理论和应用领域都将是深远的。
英文摘要：
Statistical inference from data is a foundational task in science. Recently, it has received growing attention for its central role in inference systems of primary interest in data sciences and machine learning. However, the understanding of statistical inference is not that solid while remains as a matter of subjective belief or as the routine procedures once claimed objective. We here show that there is an objective description of statistical inference for long sequence of exchangeable binary random variables, the prototypal stochasticity in theories and applications. A linear differential equation is derived from the identity known as de Finetti's representation theorem, and it turns out that statistical inference is given by the Green's functions. Our finding is an answer to the normative issue of science that pursues the objectivity based on data, and its significance will be far-reaching in most pure and applied fields.
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