  We study several bayesian inference problems for irreversible stochastic
epidemic models on networks from a statistical physics viewpoint. We derive
equations which allow 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. Our method, which is
based on the Belief Propagation algorithm, is efficient, naturally distributed,
and exact on trees. As a particular case, we consider the problem of finding
the "zero patient" of a SIR or SI epidemic given a snapshot of the state of the
network at a later unknown time. Numerical simulations show that our method
outperforms previous ones on both synthetic and real networks, often by a very
large margin.
