Inferring Microscopic Explanatory Structures from Observational Constraints via Large Deviations

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.

💡 Research Summary

The paper investigates how macroscopic observational constraints delimit the set of admissible microscopic explanatory structures when no intrinsic order, dynamics, or causal relations are assumed a priori. Starting from an unordered collection of measurement outcomes, the authors formulate inference as a constrained large‑deviation problem. They first define a reference probability measure π on the space of microscopic labels X that reflects only the symmetry of the measurement apparatus: for each label x, π(x) is proportional to the inverse size of its orbit under the transformation group G associated with the measurement setup. This reference measure is independent of any prior knowledge about the underlying system and serves as the baseline distribution Q(ω)=∏π(x_i) for a multiset ω of N outcomes, where permutations of indices are identified (i.e., ω belongs to X^N/S_N).

Macroscopic observables are introduced as expectation constraints of the form ⟨M(ω,σ)⟩_P/N = m, where M is an extensive function that depends on the unordered data ω and a hypothetical ordering σ∈S_N. The ordering σ does not correspond to any observable temporal or causal sequence; it merely labels a possible relational structure (e.g., a permutation that would impose an order if one were to be hypothesized). The constraint must be invariant under both the measurement‑reference group G and the permutation group S_N, ensuring that the macroscopic quantity does not depend on arbitrary relabelings.

To incorporate the macroscopic constraint while staying as close as possible to the reference distribution, the authors minimize the Kullback‑Leibler divergence D(P‖Q) subject to the constraint. The variational problem yields a family of tilted distributions \