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

arXiv: https://arxiv.org/abs/2504.13927 · HTML: https://mlsanity.com/paper/2504.13927/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 30 references that are in the map.
- First posted on arXiv 14 Apr 2025.

## Builds on (most cited first, 1 of 1)

- [Statistical Physics of Deep Neural Networks: Generalization Capability, Beyond the Infinite Width, and Feature Learning](https://mlsanity.com/paper/2501.19281/) (2025)

## Built on by (most cited first, 1 of 1)

- [Data-Driven Energy-Based Learning via Gibbs Measures on Hierarchical Structures](https://mlsanity.com/paper/2606.30064/) (2026)

## Abstract (arXiv)

In this paper, we investigate tree-indexed Markov chains (Gibbs measures) defined by a Hamiltonian that couples two Ising layers: hidden spins (s(x) ∈ ± 1) and observed spins (σ(x) ∈ ± 1) on a Cayley tree. The Hamiltonian incorporates Ising interactions within each layer and site-wise emission couplings between layers, extending hidden Markov models to a bilayer Markov random field. 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.