Bayesian Inference with Structured Signal: Static Replica Symmetry Breaking on the Nishimori Line in the Planted Spin Glass
2026-08-03 • Information Theory
Information Theory
AI summaryⓘ
The authors studied how correlations within a signal affect the ability to infer it correctly, using a simple model based on spin glasses. They found that in one phase, having correlations makes it easier to recover the signal, while in another phase, the signal's structure already gives some information on its own. They also discovered a complex phase called replica symmetry breaking (RSB) that appears in the posterior distribution, which affects inference methods like Belief Propagation. Their work highlights how the internal structure of the signal can change the difficulty of Bayesian inference.
Bayesian inferencesignal priorplanted spin glassrandom regular graphsIsing modelparamagnetic regimeferromagnetic regimereplica symmetry breaking (RSB)Nishimori conditionsBelief Propagation
Authors
Andrea Vincenzo Dell'Abate, Louise Budzynski
Abstract
A common assumption in theoretical models of Bayesian inference is that the signal has i.i.d. components. To study the effect of correlations in the signal prior, we consider a minimal model: the planted spin glass on random regular graphs, where the signal is sampled from an Ising model with coupling $κ$. Depending on the phase of the prior, we find that adding structure in the signal can either help or hinder inference. In the paramagnetic regime, correlations in the signal lower the reconstruction threshold, so that weaker signal strength is sufficient for recovery. In the ferromagnetic regime, the prior alone already enables partial recovery, and we identify the threshold above which the observations provide additional information. When the prior itself is in a replica symmetry breaking (RSB) phase, we detect a static RSB transition in the posterior under Nishimori conditions. This provides an example where a non-separable, correlated prior leads to static RSB in a Bayes-optimal inference problem. We discuss the consequences of this glassy phase for algorithmic performance, in particular for Belief Propagation.