An Oracle Approach for Interaction Neighborhood Estimation in Random Fields

TL;DR

This paper introduces a new model selection method for estimating interaction neighborhoods in random fields, demonstrating its theoretical guarantees and practical efficiency, especially in Ising models.

Contribution

It proposes an oracle-based selection rule for neighborhood estimation that is applicable to a broad class of models, including Ising models without restrictions.

Findings

The selection rule satisfies an oracle inequality.

The method is computationally efficient for Ising models.

Practical effectiveness shown through simulation studies.

Abstract

We consider the problem of interaction neighborhood estimation from the partial observation of a finite number of realizations of a random field. We introduce a model selection rule to choose estimators of conditional probabilities among natural candidates. Our main result is an oracle inequality satisfied by the resulting estimator. We use then this selection rule in a two-step procedure to evaluate the interacting neighborhoods. The selection rule selects a small prior set of possible interacting points and a cutting step remove from this prior set the irrelevant points. We also prove that the Ising models satisfy the assumptions of the main theorems, without restrictions on the temperature, on the structure of the interacting graph or on the range of the interactions. It provides therefore a large class of applications for our results. We give a computationally efficient procedure in these models. We finally show the practical efficiency of our approach in a simulation study.