Small-Coupling Dynamic Cavity: a Bayesian mean-field framework for epidemic inference

We present the Small-Coupling Dynamic Cavity (SCDC) method, a novel generalized mean-field approximation for epidemic inference and risk assessment within a fully Bayesian framework. SCDC accounts for non-causal effects of observations and uses a graphical model representation of epidemic processes to derive self-consistent equations for edge probability marginals. A small-coupling expansion yields time-dependent cavity messages capturing individual infection probabilities and observational conditioning. With linear computational cost per iteration in the epidemic duration, SCDC is particularly efficient and valid even for recurrent epidemic processes, where standard methods are exponentially complex. Tested on synthetic networks, it matches Belief Propagation in accuracy and outperforms individual-based mean-field methods. Notably, despite being derived as a small-infectiousness expansion, SCDC maintains good accuracy even for relatively large infection probabilities. While convergence issues may arise on graphs with long-range correlations, SCDC reliably estimates risk. Future extensions include non-Markovian models and higher-order terms in the dynamic cavity framework.