Inferring Microscopic Explanatory Structures from Observational Constraints via Large Deviations

TL;DR

This work addresses how macroscopic observational constraints restrict admissible microscopic explanatory structures when no intrinsic order is assumed. It adopts a constrained large-deviation approach, minimizing $D(P\|Q)$ with a reference measure $Q$ fixed by the measurement setup, and selects the hypothesis $\sigma$ that makes the realized observation most typical under $P^*_\sigma$. The key contribution is a principled mechanism by which relational structures (e.g., order) can emerge as explanations, or be rejected, purely from inference under constraint, even when the macroscopic constraint is symmetry-invariant. The authors illustrate the mechanism with a minimal binary model, showing how macroscopic data can induce symmetry breaking in explanations while leaving the observational constraint symmetric, highlighting underdetermination and the selective power of large-deviation reasoning in relational inference.

Abstract

We study how macroscopic observational constraints restrict admissible microscopic explanatory structures when no intrinsic order or dynamics is assumed a priori. Starting from an unordered collection of measurement outcomes, we formulate inference as a constrained large deviation problem, selecting probability assignments that minimize relative entropy with respect to a reference measure determined solely by the measurement setup. We show that among all microscopic structures compatible with a given macroscopic constraint, those rendering the observation statistically most typical are selected. As an explicit illustration, we demonstrate how ordered microscopic structures can emerge purely from inference under constraint, even when the reference measure is fully permutation symmetric. Order is thus not assumed but inferred, serving here only as an illustrative example of a broader class of relational explanatory hypotheses constrained by observation.