Dynamics of epidemics from cavity master equations: Susceptible-infectious-susceptible models.

We apply the recently introduced cavity master equation (CME) to epidemic models and compare it to previously known approaches. We show that CME seems to be the formal way to derive (and correct) dynamic message passing (rDMP) equations that were previously introduced in an intuitive ad hoc manner. CME outperforms rDMP in all cases studied.

Gauge-free cluster variational method by maximal messages and moment matching.

We present an implementation of the cluster variational method (CVM) as a message passing algorithm. The kind of message passing algorithm used for CVM, usually named generalized belief propagation (GBP), is a generalization of the belief propagation algorithm in the same way that CVM is a generalization of the Bethe approximation for estimating the partition function. However, the connection between fixed points of GBP and the extremal points of the CVM free energy is usually not a one-to-one correspondence because of the existence of a gauge transformation involving the GBP messages.

Bayesian inference of epidemics on networks via belief propagation.

We study several Bayesian inference problems for irreversible stochastic epidemic models on networks from a statistical physics viewpoint. We derive equations which allow us to accurately compute the posterior distribution of the time evolution of the state of each node given some observations. At difference with most existing methods, we allow very general observation models, including unobserved nodes, state observations made at different or unknown times, and observations of infection times, possibly mixed together.

Inference algorithm for finite-dimensional spin glasses: belief propagation on the dual lattice.

Starting from a cluster variational method, and inspired by the correctness of the paramagnetic ansatz [at high temperatures in general, and at any temperature in the two-dimensional (2D) Edwards-Anderson (EA) model] we propose a message-passing algorithm--the dual algorithm--to estimate the marginal probabilities of spin glasses on finite-dimensional lattices. We use the EA models in 2D and 3D as benchmarks. The dual algorithm improves the Bethe approximation, and we show that in a wide range of temperatures (compared to the Bethe critical temperature) our algorithm compares very well with Monte Carlo simulations, with the double-loop algorithm, and with exact calculation of the ground state of 2D systems with bimodal and Gaussian interactions.