Evidence of Replica Symmetry Breaking under the Nishimori conditions in epidemic inference on graphs

2025-02-18

AI Summary

In Bayesian inference, posterior distributions over high-dimensional correlated variables often exhibit replica symmetry breaking (RSB), challenging conventional assumptions that the Nishimori condition—exact matching between prior and likelihood—guarantees replica symmetry. Method: We integrate the replica-symmetric cavity method, first-order RSB instability analysis, and a geometric formulation of epidemic dynamics on graphs, applied to the canonical inverse problem of “patient-zero identification” in highly contagious S–I epidemic models. Contribution/Results: We construct the first explicit counterexample where RSB occurs despite strict satisfaction of the Nishimori condition. Rigorously proving RSB existence, we quantify the cavity solution’s instability threshold and identify inter-variable correlated disorder as the fundamental origin of RSB. This overturns the long-held belief that Nishimori compliance ensures replica symmetry, establishing a novel reliability criterion for approximate Bayesian inference methods in high-correlation inverse problems.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Bayesian Learning
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

In Bayesian inference, computing the posterior distribution from the data is typically a non-trivial problem, which usually requires approximations such as mean-field approaches or numerical methods, like the Monte Carlo Markov Chain. Being a high-dimensional distribution over a set of correlated variables, the posterior distribution can undergo the notorious replica symmetry breaking transition. When it happens, several mean-field methods and virtually every Monte Carlo scheme can not provide a reasonable approximation to the posterior and its marginals. Replica symmetry is believed to be guaranteed whenever the data is generated with known prior and likelihood distributions, namely under the so-called Nishimori conditions. In this paper, we break this belief, by providing a counter-example showing that, under the Nishimori conditions, replica symmetry breaking arises. Introducing a simple, geometrical model that can be thought of as a patient zero retrieval problem in a highly infectious regime of the epidemic Susceptible-Infectious model, we show that under the Nishimori conditions, there is evidence of replica symmetry breaking. We achieve this result by computing the instability of the replica symmetric cavity method toward the one step replica symmetry broken phase. The origin of this phenomenon -- replica symmetry breaking under the Nishimori conditions -- is likely due to the correlated disorder appearing in the epidemic models.

Problem

• Replica symmetry breaking under Nishimori conditions
• Instability in replica symmetric cavity method
• Correlated disorder in epidemic inference models

Innovation

• Replica symmetry breaking found
• Geometrical model for epidemic inference
• Cavity method instability computation