Evidence of replica symmetry breaking under the Nishimori conditions in epidemic inference on graphs

In Bayesian inference, computing the posterior distribution from the data is typically a nontrivial 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 that happens, several mean-field methods and virtually every Monte Carlo scheme cannot provide a reasonable approximation to the posterior and its marginals. Replica symmetry is believed to be guaranteed whenever the data are generated with known prior and likelihood distributions, namely under the so-called Nishimori conditions. In this paper, we break this belief by providing a counterexample 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.