Ising Models with Hidden Markov Structure: Applications to Probabilistic Inference in Machine Learning

In this paper, we investigate a Hamiltonian that incorporates Ising interactions between hidden $\pm 1$ spins, alongside a data-dependent term that couples the hidden and observed variables. Specifically, we explore translation-invariant Gibbs measures (TIGM) of this Hamiltonian on Cayley trees. Under certain explicit conditions on the model's parameters, we demonstrate that there can be up to three distinct TIGMs. Each of these measures represents an equilibrium state of the spin system. These measures provide a structured approach to inference on hierarchical data in machine learning. They have practical applications in tasks such as denoising, weakly supervised learning, and anomaly detection. The Cayley tree structure is particularly advantageous for exact inference due to its tractability.