# Interaction Order Estimation in Tensor Curie–Weiss Models

## TL;DR

This paper studies the estimation of interaction parameters in Curie–Weiss and Ising models, showing when it is possible or impossible to estimate these parameters based on temperature and model structure.

## Contribution

The paper introduces a threshold function β*(p) that determines the feasibility of estimating the interaction order p in Curie–Weiss models.

## Key findings

- Joint estimation of β and p is impossible due to contiguity.
- Estimation of p is impossible if β is unknown.
- Consistent estimation of p is possible only when β exceeds β*(p).

## Abstract

In this paper, we consider the problem of estimating the interaction parameter p of a p-spin Curie–Weiss model at inverse temperature β, given a single observation from this model. We show, by a contiguity argument, that joint estimation of the parameters β and p is impossible, which implies that the estimation of p is impossible if β is unknown. These impossibility results are also extended to the more general p-spin Erdös–Rényi Ising model. The situation is more delicate when β is known. In this case, we show that there exists an increasing threshold function β*(p), such that for all β, consistent estimation of p is impossible when β*(p)>β, and for almost all
β, consistent estimation of p is possible for β*(p)<β.