Parametrized Families of Gibbs Measures and Their Statistical Inference

M. Denker, M. Kesseböhmer, A. O. Lopes, S. R. C. Lopes

correctmedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The candidate conflates two different likelihoods. For the non-invariant eigenmeasure νθ used in the paper, log νθ([x0,…,xn−1]) is not the exact exponential-family form S_nF_θ(x)−nP(F_θ); rather, it equals an integral ∫ e^{−nP(F_θ)+S_nF_θ(cx)} dν_θ(x), and its score/Hessian carry non-negligible O(1) and random terms. The paper’s Lemmas 3.3–3.4 compute these derivatives precisely and lead to the non-Gaussian limit √n(θ̂_n−θ) ⇒ G^{-1}(N)N^t (Theorem 2.8), including a 1D illustration N/(N^2−σ^2). The model’s “exact exponential-family identity,” deterministic observed information, and Σ_{μ_θ}^{−1}N limit are therefore incorrect for the νθ-based MLE. However, we note a likely flaw in the paper’s Section 7.2 claiming the same G^{-1}(N)N^t limit for the maximum potential estimator (MPE): the MPE solves ∇P(F_{θ̃_n})=S_nF/n and, by the usual delta method with Hess P(F_θ)=Σ_{μ_θ}, should be asymptotically normal with covariance Σ_{μ_θ}^{−1}. This does not affect the main MLE result but deserves revision.

Referee report (LaTeX)

\textbf{Recommendation:} major revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The main result for the ν-based MLE is well-motivated and, in my view, correct: the derivative expansions and mixing-based limit theory substantiate the non-Gaussian G\^{-1}(N)N\^t limit and are illustrated by a clear one-dimensional example. The manuscript should, however, carefully distinguish the ν-MLE from the maximum potential estimator (MPE). As written, Section 7.2 claims the same limit for the MPE; this likely conflicts with the delta-method normal limit for the implicit equation ∇P(F\_{θ̃})=S\_nF/n with Hess P=Σ\_μ. Clarifying or correcting this point is important for accuracy.