Hybrid Geometry-Adaptive MCMC for Bayesian Inference in Higher-Order Ising Models

We address the inverse problem for the mean-field Ising model with two- and three-body interactions using a Bayesian framework. Parameter recovery in this setting is notoriously difficult, particularly near phase transitions, at criticality, and under non-identifiability, where conventional estimators and standard MCMC samplers fail. To overcome these challenges, we develop a hybrid algorithm that combines Adaptive Metropolis Hastings with geometry-aware Riemannian manifold Hamiltonian dynamics. This approach yields substantially improved mixing and convergence in the three-dimensional parameter space. Through simulated experiments across representative regimes, we demonstrate that the method achieves accurate density reconstruction and reliable uncertainty quantification even in settings where existing approaches are unstable or inapplicable.