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Level sets and maximum likelihood estimation for the Ising model

Tomasz Skalski, Tomasz Stroiński

TL;DR

This work advances the understanding of when the maximum likelihood estimator exists in discrete exponential families by leveraging the set-of-uniqueness framework and applying it to the Ising model on a complete graph. The authors give a full characterization for the existence of the MLE among level sets of the Ising model, achieved through a geometric reduction that maps level sets to planar polygons and uses convexity to determine uniqueness. They also derive new bounds for the size of the smallest set of uniqueness for products of Rademacher functions, including explicit constructions and asymptotic behavior across parameters $k$ and $q$, and propose a precise conjecture for the case $q=2$. The results connect MLE existence criteria with combinatorial geometry on the hypercube, informing both statistical estimation and subcube-intersection problems with potential applications to hypercube processors and Ising-type models.

Abstract

Bogdan et al. established a new criterion to determine the existence of a maximum likelihood estimator in discrete exponential families. It uses the notion of the set of uniqueness, which allows to apply the problem to the Ising model from statistical mechanics. We propose a full characterization of the existence of the MLE in the Ising model among the level sets used in related combinatorial problems. Then we establish new bounds for the size of the smallest set of uniqueness for the products of Rademacher functions.

Level sets and maximum likelihood estimation for the Ising model

TL;DR

This work advances the understanding of when the maximum likelihood estimator exists in discrete exponential families by leveraging the set-of-uniqueness framework and applying it to the Ising model on a complete graph. The authors give a full characterization for the existence of the MLE among level sets of the Ising model, achieved through a geometric reduction that maps level sets to planar polygons and uses convexity to determine uniqueness. They also derive new bounds for the size of the smallest set of uniqueness for products of Rademacher functions, including explicit constructions and asymptotic behavior across parameters and , and propose a precise conjecture for the case . The results connect MLE existence criteria with combinatorial geometry on the hypercube, informing both statistical estimation and subcube-intersection problems with potential applications to hypercube processors and Ising-type models.

Abstract

Bogdan et al. established a new criterion to determine the existence of a maximum likelihood estimator in discrete exponential families. It uses the notion of the set of uniqueness, which allows to apply the problem to the Ising model from statistical mechanics. We propose a full characterization of the existence of the MLE in the Ising model among the level sets used in related combinatorial problems. Then we establish new bounds for the size of the smallest set of uniqueness for the products of Rademacher functions.

Paper Structure

This paper contains 12 sections, 16 theorems, 50 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Discrete exponential families
  3. Rademacher functions
  4. Ising model
  5. Level sets
  6. Introduction and motivation
  7. Full characteristics for the Ising model
  8. Extremal properties
  9. Partial results in general case
  10. Proofs
  11. Main characterization
  12. Proofs of Lemmas

Key Result

Lemma 2.2

Let $\varphi\in\mathcal{B}^k_q$ and $D\subset\{0,1,\ldots,k\}$. If $U=W_{D}$, then where $T_j = \sum_{S\in S_j} \alpha_S$ for every $j=0,1\,\ldots,k$ and $\alpha_S\in\mathbb{R}$ for every $S\in\mathcal{S}^k_q = S_0 \cup S_1 \cup \ldots \cup S_q.$

Figures (2)

  • Figure 1: Illustration of the Lemma \ref{['lem:symmetry']}.
  • Figure 2: Illustration of the Corollary \ref{['cor:incr']}.

Theorems & Definitions (37)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • Remark 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 27 more