Title: Estimation in tensor Ising models
Authors: Somabha Mukherjee - University of Pennsylvania (United States)
Jaesung Son - Columbia University (United States)
Bhaswar Bhattacharya - University of Pennsylvania (United States) [presenting]
Abstract: The $p$-tensor Ising model is a one-parameter discrete exponential family for modeling dependent binary data, where the sufficient statistic is a multi-linear form of degree $p \geq 2$. This is a natural generalization of the matrix Ising model that provides a convenient mathematical framework for capturing higher-order dependencies in complex relational data. We will discuss the problem of estimating the natural parameter of the $p$-tensor Ising model given a single sample from the distribution on $N$ nodes. Our estimate is based on the maximum pseudo-likelihood (MPL) method, which provides a computationally efficient algorithm for estimating the parameter that avoids computing the intractable partition function. General conditions under which the MPL estimate is $\sqrt N$-consistent will be presented. Our conditions are robust enough to handle a variety of commonly used tensor Ising models, including models where the rate of estimation undergoes a phase transition, such as the well-known stochastic block model. Finally, we will discuss the precise fluctuations of the MPL estimate in the special case of the Curie-Weiss model. The MPL estimate saturates the Cramer-Rao lower bound at all points above the estimation threshold, that is, the MPL estimate incurs no loss in asymptotic efficiency, even though it is obtained by minimizing only an approximation of the true likelihood function for computational tractability.