Bayesian Inverse Ising Problem with Three-body Interactions

2024-04-08

AI Summary

This work addresses the Bayesian parameter inference challenge for higher-order (three-body) Ising models near phase-transition critical points and in parameter non-identifiability regions—where the partition function is analytically intractable and conventional MCMC methods suffer from slow convergence and poor mixing. Methodologically, we propose a novel hybrid Bayesian inference framework: (1) an extensible mean-field approximation tailored to three-body interactions, generalizing beyond pairwise couplings; and (2) the first unified, adaptive sampler integrating Adaptive Metropolis–Hastings (AMH), Hamiltonian Monte Carlo (HMC), and Manifold-corrected Metropolis-Adjusted Langevin Algorithm (MALA). Experiments demonstrate that our approach maintains rapid convergence and strong sampling efficiency near criticality, significantly improving accuracy and robustness in estimating three-body coupling parameters. This framework establishes a new computational paradigm for Bayesian inference in higher-order statistical physical models.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Probabilistic Inference
• Search and Optimization: Sampling/Simulation-based Search

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• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
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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.

Problem

• Solving inverse problems for Ising models with multi-body interactions
• Overcoming parameter recovery difficulties near phase transitions
• Developing hybrid MCMC for improved sampling in complex parameter spaces

Innovation

• Hybrid algorithm combines adaptive and geometry-aware MCMC
• Uses Riemannian manifold Hamiltonian dynamics for sampling
• Improves mixing in three-dimensional parameter space