[Paper Review] Inferring Microscopic Explanatory Structures from Observational Constraints via Large Deviations

Akihisa Ichiki|arXiv (Cornell University)|Feb 6, 2026

Philosophy and History of Science0 citations

TL;DR

The paper formulates inference of microscopic explanatory structures from unordered observational data as a constrained large deviation problem, selecting the most typical explanations under macroscopic constraints.

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.

Motivation & Objective

• Motivate how macroscopic observational constraints restrict admissible microscopic explanations without assuming intrinsic order or dynamics.
• Propose a formal framework that uses a measurement-induced reference measure and a constrained large deviation principle to select plausible microscopic structures.
• Show that relational structures like order can emerge as explanatory hypotheses from constraint-driven inference.
• Illustrate the mechanism with a minimal binary model to demonstrate symmetry preservation and potential symmetry breaking at the explanatory level.

Proposed method

• Define an unordered collection of measurement outcomes and construct a measurement-induced reference measure  on labels using the symmetry group action of the measurement setup.
• Introduce hypothetical relational structures via an ordering   sigma and a macroscopic observable M(  , sigma) that depends on ordering, with the macroscopic constraint requiring the expectation  M( , sigma)/N = m to be invariant under permutations and gauge transformations.
• Apply constrained large deviation theory by minimizing D(P||Q) subject to the constraint, yielding P*_{ , sigma}()   proportional to Q() exp(lambda M( , sigma)).
• Use Sanov's theorem to compare realized observations to hypothesized structures, selecting sigma* that maximizes P*_{ , sigma}(_{ ext{obs}}).
• Demonstrate a minimal binary model with binary labels and a permutation symmetry to show how macroscopic constraints can rule out or select microscopic hypotheses.

Experimental results

Research questions

• RQ1 Can relational explanatory structures (like order) be inferred from unordered observations without assuming intrinsic organization?
• RQ2 How do macroscopic constraints influence the selection among possible microscopic explanatory structures under a symmetry-preserving observational framework?
• RQ3 Under what conditions does a symmetry-breaking explanatory structure emerge as the most typical explanation given the constraint and reference measure?
• RQ4 What is the role of large deviation principles in selecting explanations rather than dynamics or sequential data?
• RQ5 How does a minimal binary model illustrate the mechanism of constraint-driven hypothesis selection?

Key findings

• Relational structures such as order can emerge as explanatory hypotheses solely from constraint-driven inference, even when the reference measure is symmetric.
• Macroscopic constraints are required to be invariant under permutations and gauge transformations, while the selected explanatory structure need not be.
• The selection principle is effectively a maximum-likelihood-like procedure over relational structures, induced by a variational large deviation framework.
• Macroscopic observations can rule out entire classes of microscopic models, not just individual hypotheses, via the large deviation mechanism.
• A minimal binary model demonstrates degeneration, rejection, or selection of microscopic hypotheses depending on observed macroscopic values.