We propose an extension to the functional modelling methods described by Dawid and Stone (1982 Ann. Stat. 10 1119-38) that leads naturally to a method for selecting vague parameter priors for Bayesian analyses involving stochastic population models. Motivated by applications from quantum optics and epidemiology, we focus on analysing observed sequences of event times obeying a non-homogeneous Poisson process, although the techniques are more widely applicable. The extended functional modelling approach is illustrated for the particular case of Bayesian estimation of the death rate in the immigration-death model from observation of the death times only. It is shown that the prior selected naturally leads to a well defined posterior density for parameters and avoids some undesirable pathologies reported by Gibson and Renshaw (2001a Inverse Problems 17 455-66, 2001b Stat. Comput. 11 347-58) for the case of exponential priors. Some limitations of the approach are also discussed.