We consider the problem of Bayesian inference for infection rates in a multi-type stochastic epidemic model in which the population has a given structure, given data on final outcome. For such data, a likelihood is both analytically and numerically intractable. This problem can be overcome by imputation of suitable latent variables. We describe two such approaches based on different representations of the epidemic model. We also consider extentions to the methodology for the situation where the observed data are a fraction of the entire population. The methods are illustrated with data on influenza outbreaks.