Efficient Estimation in Tensor Ising Models

The tensor Ising model is a discrete exponential family used for modeling binary data on networks with not just pairwise, but higher-order dependencies. A particularly important class of tensor Ising models are the tensor Curie-Weiss models, where all tuples of nodes of a particular order interact with the same intensity. The maximum likelihood estimator (MLE) is not explicit in this model, due to the presence of an intractable normalizing constant in the likelihood, and a computationally efficient alternative is to use the maximum pseudolikelihood estimator (MPLE). In this paper, we show that the MPLE is in fact as efficient as the MLE (in the Bahadur sense) in the $2$-spin model, and for all values of the null parameter above $\log 2$ in higher-order tensor models. Even if the null parameter happens to lie within the very small window between the threshold and $\log 2$, they are equally efficient unless the alternative parameter is large. Therefore, not only is the MPLE computationally preferable to the MLE, but also theoretically as efficient as the MLE over most of the parameter space. Our results extend to the more general class of Erd\H{o}s-R\'enyi hypergraph Ising models, under slight sparsities too.