Bernoulli Society Journal Lecture
Conference
Category: Bernoulli Society for Mathematical Statistics and Probability (BS)
Abstract
The $p$-spin Curie-Weiss model, for $p\geq 3$, is a binary spin-system where all possible $p$-tuples of interactions are present. It corresponds to the tensor Ising model on the complete $p$-uniform hypergraph and is a prototypical example of a mean-field spin system with higher-order interactions. In this talk we will discuss the fluctuations (asymptotic distribution) of the maximum likelihood (ML) estimates of the inverse temperature and the magnetic field given a single realization from the $p$-spin Curie-Weiss model. Several new phase transitions and surprising limit theorems will emerge, such as the existence of a ‘critical’ curve in the parameter space, where the limiting distributions of the ML estimates are mixtures with both continuous and discrete components, and the existence of certain ‘special’ points in the parameter space where the ML estimates exhibit a superefficiency phenomenon, converging to a non-Gaussian limiting distribution at a rate faster than the usual parametric rate. Using these results, we can obtain asymptotically valid confidence intervals for the inverse temperature and the magnetic field at all points in the parameter space where consistent estimation is possible. (Joint work with Somabha Mukherjee, and Jaesung Son.)