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This thesis investigates the representation of a stochastic epidemic process as a directed random graph; we use this representation to impute the missing information in final size data to make Bayesian statistical inference about the model parameters using MCMC techniques. The directed random graph representation is analysed, in particular its behaviour under the condition that the epidemic has a given final size. This is used to construct efficient updates for MCMC algorithms. The MCMC method is extended to include two-level mixing models and two-type models, with a general framework given for an arbitrary number of levels and types. Partially observed epidemics, that is, where the number of susceptibles is unknown or where only a subset of the population is observed, are analysed. The method is applied to several well known data sets and comparisons are made with previous results. Finally, the method is applied to data of an outbreak of Equine Influenza (H3N8) at Newmarket in 2003, with a comparison to another analysis of the same data. Practical issues of implementing the method are discussed and are overcome using parallel computing (GNU OpenMP) and arbitrary precision arithmetic (GNU MPFR).