Large Deviation Strategy for Inverse Problem

Abstract

Taken traditionally as a no-go theorem against the theorization of inductive processes, Duhem-Quine thesis may interfere with the essence of statistical inference. This difficulty can be resolved by Micro-Macro duality \cite{Oj03, Oj05} which clarifies the importance of specifying the pertinent aspects and accuracy relevant to concrete contexts of scientific discussions and which ensures the matching between what to be described and what to describe in the form of the validity of duality relations. This consolidates the foundations of the inverse problem, induction method, and statistical inference crucial for the sound relations between theory and experiments. To achieve the purpose, we propose here Large Deviation Strategy (LDS for short) on the basis of Micro-Macro duality, quadrality scheme, and large deviation principle. According to the quadrality scheme emphasizing the basic roles played by the dynamics, algebra of observables together with its representations and universal notion of classifying space, LDS consists of four levels and we discuss its first and second levels in detail, aiming at establishing statistical inference concerning observables and states. By efficient use of the central measure, we will establish a quantum version of Sanov's theorem, the Bayesian escort predictive state and the widely applicable information criteria for quantum states in LDS second level. Finally, these results are reexamined in the context of quantum estimation theory, and organized as quantum model selection, i.e., a quantum version of model selection.