We consider posterior sampling in situations where data are incomplete in such a way that likelihood functions corresponding to portions of the data must be constructed. Slice sampling, a recently developed Markov chain Monte Carlo method, makes such computation feasible. Such situations arise in the context of stochastic networks where an overall predictive waiting time is comprised of convolutions and finite mixtures of individual transitions and portions of the individual transition information is unobserved for some records. We present applications involving flowgraph models in the reliability setting; however, the methodology presented is relevant to any application where likelihood functions cannot be specified in closed form but can be evaluated numerically. Copyright © 2006 John Wiley & Sons, Ltd.