Title: Bayesian Inverse Ising Problem with Three-body Interactions

Abstract: In this paper, we solve the inverse Ising problem with three-body interaction. Using the mean-field approximation, we find a tractable expansion of the normalizing constant. This facilitates estimation, which is known to be quite challenging for the Ising model. We then develop a novel hybrid MCMC algorithm that integrates Adaptive Metropolis Hastings (AMH), Hamiltonian Monte Carlo (HMC), and the Manifold-Adjusted Langevin Algorithm (MALA), which converges quickly and mixes well. We demonstrate the robustness of our algorithm using data simulated with a structure under which parameter estimation is known to be challenging, such as in the presence of a phase transition and at the critical point of the system.