Open Access
April 2020 Joint estimation of parameters in Ising model
Promit Ghosal, Sumit Mukherjee
Ann. Statist. 48(2): 785-810 (April 2020). DOI: 10.1214/19-AOS1822


We study joint estimation of the inverse temperature and magnetization parameters $(\beta ,B)$ of an Ising model with a nonnegative 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 pseudo-likelihood 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 Erdős–Renyi with parameter $p>0$ independent of $n$, thus confirming that estimation is harder on approximately regular graphs with large degree.


Download Citation

Promit Ghosal. Sumit Mukherjee. "Joint estimation of parameters in Ising model." Ann. Statist. 48 (2) 785 - 810, April 2020.


Received: 1 February 2018; Revised: 1 October 2018; Published: April 2020
First available in Project Euclid: 26 May 2020

zbMATH: 07241569
MathSciNet: MR4102676
Digital Object Identifier: 10.1214/19-AOS1822

Primary: 62F12
Secondary: 60F10

Keywords: Ising model , pseudo-likelihood

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 2 • April 2020
Back to Top