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

## Abstract

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.

## Citation

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

## Information

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

Subjects:
Primary: 62F12
Secondary: 60F10

Keywords: Ising model , pseudo-likelihood
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