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Joint Estimation in Potts Model

Somabha Mukherjee, Sumit Mukherjee, Sayar Karmakar

Abstract

In this paper, we study estimation of parameters in a two-parameter Potts model with $q$ colors and coupling matrix $A_N$. We characterize concrete sufficient conditions for existence of the pseudo-likelihood estimator of the Potts model, in terms of the local magnetic fields, and give sufficient conditions for the validity of the above characterization. We then provide sufficient criteria for estimation of both parameters at the optimal rate $\sqrt{N}$. In particular, if $A_N$ is the scaled adjacency matrix of a graph $G_N$, then we show that joint estimation is possible if either $G_N$ has bounded degree or is irregular. In contrast, we give an example of a graph sequence $G_N$ which is approximately regular and dense, where no consistent estimator exists. We also show that one-parameter estimation at the optimal rate $\sqrt{N}$ holds under much milder conditions when the other parameter is known. Along the way, we develop a concentration result for mean-field Potts models using the framework of nonlinear large deviations. Compared to the Ising case, our results for the Potts case require a novel analysis across multiple colors.

Joint Estimation in Potts Model

Abstract

In this paper, we study estimation of parameters in a two-parameter Potts model with colors and coupling matrix . We characterize concrete sufficient conditions for existence of the pseudo-likelihood estimator of the Potts model, in terms of the local magnetic fields, and give sufficient conditions for the validity of the above characterization. We then provide sufficient criteria for estimation of both parameters at the optimal rate . In particular, if is the scaled adjacency matrix of a graph , then we show that joint estimation is possible if either has bounded degree or is irregular. In contrast, we give an example of a graph sequence which is approximately regular and dense, where no consistent estimator exists. We also show that one-parameter estimation at the optimal rate holds under much milder conditions when the other parameter is known. Along the way, we develop a concentration result for mean-field Potts models using the framework of nonlinear large deviations. Compared to the Ising case, our results for the Potts case require a novel analysis across multiple colors.

Paper Structure

This paper contains 20 sections, 19 theorems, 301 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Our contributions
  4. Main Results
  5. Non mean-field interactions
  6. Irregular interactions
  7. Future directions
  8. Outline of the paper
  9. Proof of Theorem \ref{['thm:theorem1']}
  10. Numerical Study
  11. Acknowledgement
  12. Proof of Theorem \ref{['bdgr']}
  13. Proof of Theorem \ref{['irrgr']}
  14. Proof of Theorem \ref{['jinest']}
  15. Proof of Theorem \ref{['partialestm']}
  16. ...and 5 more sections

Key Result

Theorem 1.1

Suppose $\bm X$ is a sample from the Potts model eq:pmf, where the coupling matrix $\bm A_N$ has non-negative entries, and satisfies conditions as1 and as2. If $(\beta,\bm B) \in \Theta := (0,\infty)\times \mathbb{R}^{q-1}$, then the following conclusions hold: $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure : (a)
  • Figure : (a)
  • Figure : (b)

Theorems & Definitions (34)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Lemma A.1
  • Proposition E.1
  • proof
  • ...and 24 more