A Bayesian framework for the ratio of two Poisson rates in the context of vaccine efficacy trials

In many applications, we assume that two random observations x and y are generated
according to independent Poisson distributions and we are interested in performing
statistical inference on the ratio of the two incidence rates, called the relative risk in vaccine efficacy trials, in which context x and y are the numbers of cases in the vaccine and the control groups respectively. In this paper we start by defining a natural semi-conjugate family of prior distributions for this model, allowing straightforward computation of the posterior inference. Following theory on reference priors, we define the reference prior for the partial immunity model when the relative risk is the parameter of interest. We also define a family of reference priors with partial information on the incidence rate of the unvaccinated population while remaining uninformative about the relative risk . We notice that these priors belong to the semi-conjugate family. We then demonstrate using numerical examples that Bayesian credible intervals enjoy attractive frequentist properties when using reference priors, a typical property of reference priors.