🤖 AI Summary
This study challenges the conventional assumption of independent and identically distributed signals by investigating how structured priors with correlations affect Bayesian inference performance. By constructing a planted spin glass model on random regular graphs, where signals are sampled from an Ising model with coupling strength κ, the work systematically analyzes how structured priors—spanning paramagnetic, ferromagnetic, and replica symmetry breaking (RSB) phases—influence signal reconstruction. The analysis reveals, for the first time, that non-separable correlated priors can induce a static RSB phase in the posterior distribution along the Nishimori line, thereby limiting the efficacy of algorithms such as belief propagation. Furthermore, it demonstrates that correlations in the paramagnetic regime lower the reconstruction threshold, while in the ferromagnetic regime, a critical point emerges where observations provide additional informative content.
📝 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.