Statistical inference as Green's functions

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.