Joint estimation of parameters in Ising model

TL;DR

This paper investigates the joint estimation of parameters in the Ising model, providing bounds on estimator consistency and revealing conditions under which estimation at the parametric rate is feasible or impossible.

Contribution

It introduces a general bound on the pseudolikelihood estimator's consistency and characterizes the estimation difficulty based on graph regularity and degree growth.

Findings

Estimation at rate n^{-1/2} is possible for bounded degree graphs.

Estimation remains feasible for non-regular graphs with diverging average degree.

Consistent estimation is impossible for Erdős-Rényi graphs with fixed p > 0.

Abstract

We study joint estimation of the inverse temperature and magnetization parameters $(\beta,B)$ of an Ising model with a non-negative coupling matrix $A_n$ of size $n\times n$, given one sample from the Ising model. We give a general bound on the rate of consistency of the bi-variate pseudolikelihood estimator. Using this, we show that estimation at rate $n^{-1/2}$ is always possible if $A_n$ is the adjacency matrix of a bounded degree graph. If $A_n$ is the scaled adjacency matrix of a graph whose average degree goes to $+\infty$, the situation is a bit more delicate. In this case estimation at rate $n^{-1/2}$ is still possible if the graph is not regular (in an asymptotic sense). Finally, we show that consistent estimation of both parameters is impossible if the graph is Erd\"os-Renyi with parameter $p>0$ free of $n$, thus confirming that estimation is harder on approximately regular graphs with large degree.