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

arXiv: https://arxiv.org/abs/2502.13249 · HTML: https://mlsanity.com/paper/2502.13249/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 25 references that are in the map.
- First posted on arXiv 18 Feb 2025.

## Builds on (most cited first, 5 of 5)

- [Statistical physics of inference: Thresholds and algorithms](https://mlsanity.com/paper/1511.02476/) (2015)
- [Bayes-optimal Learning of Deep Random Networks of Extensive-width](https://mlsanity.com/paper/2302.00375/) (2023)
- [Bayesian inference of epidemics on networks via Belief Propagation](https://mlsanity.com/paper/1307.6786/) (2013)
- [Bayesian reconstruction of memories stored in neural networks from their connectivity](https://mlsanity.com/paper/2105.07416/) (2021)
- [Inference in conditioned dynamics through causality restoration](https://mlsanity.com/paper/2210.10179/) (2022)

## Abstract (arXiv)

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.