Inferring hidden states in a random kinetic Ising model: replica analysis

TL;DR

This paper uses replica analysis to compute the optimal prediction error for unobserved spins in a kinetic Ising model with partial observations, providing exact results in the thermodynamic limit.

Contribution

It introduces an analytical replica-based approach to quantify the prediction error in partially observed kinetic Ising models with non symmetric couplings.

Findings

Replica method yields exact error predictions in the thermodynamic limit.

Analytical results match well with finite system simulations.

Provides insights into inference in complex spin systems.

Abstract

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.