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

TL;DR

This paper demonstrates that replica symmetry breaking can occur under Nishimori conditions in epidemic inference models, challenging the assumption that these conditions guarantee replica symmetry, with implications for inference accuracy.

Contribution

The paper provides the first counter-example showing replica symmetry breaking under Nishimori conditions in epidemic inference, using a geometrical model and cavity method analysis.

Findings

Evidence of replica symmetry breaking under Nishimori conditions

Correlated disorder in epidemic models causes symmetry breaking

Implications for inference methods in epidemic modeling

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.