Statistical inference as Green's functions
This addresses a foundational issue in science and data sciences by offering an objective approach to statistical inference, potentially impacting many fields.
The paper tackles the problem of providing an objective description of statistical inference for exchangeable binary random variables, deriving a linear differential equation from de Finetti's representation theorem and showing that inference is given by Green's functions.
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.