  We consider problems of Bayesian inference for a spatial epidemic on a graph,
where the final state of the epidemic corresponds to bond percolation, and
where only the set or number of finally infected sites is observed. We develop
appropriate Markov chain Monte Carlo algorithms, demonstrating their
effectiveness, and we study problems of optimal experimental design. In
particular, we demonstrate that for lattice-based processes an experiment on a
sparsified lattice can yield more information on model parameters than one
conducted on a complete lattice. We also prove some probabilistic results about
the behaviour of estimators associated with large infected clusters.
