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arXiv:2610.36041

Phase Transitions in Mismatched Bayesian Inference for the Ising Model

Xinyu Sun, Zhou‐Quan Wan, Shao-Kai Jian, Hong Yao

Published Oct 26, 2026• 0 citations
arXivHTMLPDF

Abstract

Bayesian inference with a mismatched model can develop an overconfident phase, characterized by posterior agreement without faithful reconstruction of the true state. We study Bayesian inference from noisy bond measurements of an Ising model in its paramagnetic phase, with true inverse temperature $β$ and measurement-noise strength $Δ$, while inference assumes corresponding parameters $β_Q$ and $Δ_Q$. Using replica field theory, we analytically show that the overconfidence phase can occur when inference underestimates the measurement noise ($Δ_Q < Δ$) or assumes a lower prior temperature than the true one ($β_Q > β$). An exact change of variables maps the problem to a correlated random-bond Ising model, where the informative, uninformative, and overconfident phases correspond to ferromagnetic, paramagnetic, and spin-glass phases, respectively. We further perform Monte Carlo simulations of the mapped model in three dimensions and establish a finite overconfident region between the informative and uninformative phases, consistent with our analytical prediction. Finally, we show that the true model parameters can be inferred directly from the measurement records, providing a natural route to calibrating the inference model.

DOI
10.48550/arxiv.2610.36041

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We consider the problem of predicting the spin states in a kinetic Ising model when spin trajectories are observed for only a finite fraction of sites. In a Bayesian setting, where the probabilistic model of the spin dynamics is assumed to be known, the optimal prediction can be computed from the conditional (posterior) distribution of unobserved spins given the observed ones. Using the replica method, we compute the error of the Bayes optimal predictor for parallel discrete time dynamics in a fully connected spin system with non-symmetric random couplings. The results, exact in the thermodynamic limit, agree very well with simulations of finite spin systems.
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We investigate the ferromagnetic-glassy transitions that separate the low-temperature ferromagnetic and spin-glass phases in the temperature-disorder phase diagram of three-dimensional Ising spin-glass models. For this purpose, we consider the cubic-lattice $\ifmmode\pm\else\textpm\fi{}J$ (Edwards-Anderson) Ising model with bond distribution $P(J)=p\ensuremath{\delta}(J\ensuremath{-}1)+(1\ensuremath{-}p)\ensuremath{\delta}(J+1)$, and present a numerical Monte Carlo study of the critical behavior along the line that marks the onset of ferromagnetism. The finite-size scaling analysis of the Monte Carlo data shows that the ferromagnetic-glassy transition line is slightly reentrant. As a consequence, for an interval of the disorder parameter ppp, around $p\ensuremath{\approx}0.77$, the system presents a low-temperature glassy phase, an intermediate ferromagnetic phase, and a high-temperature paramagnetic phase. Along the ferromagnetic-glassy transition line, magnetic correlations show a universal critical behavior with critical exponents $\ensuremath{\nu}=0.96(2)$ and $\ensuremath{\eta}=\ensuremath{-}0.39(2)$. The hyperscaling relation $\ensuremath{\beta}/\ensuremath{\nu}=(1+\ensuremath{\eta})/2$ is satisfied at the transitions, so that $\ensuremath{\beta}/\ensuremath{\nu}=0.305(10)$. This magnetic critical behavior represents a new universality class for ferromagnetic transitions in Ising-type disordered systems. Overlap correlations are apparently not critical and show a smooth behavior across the transition.
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