On implementing Jeffreys' substitution likelihood for Bayesian inference concerning the medians of unknown distributions

When statisticians are uncertain as to which parametric statistical model to use to analyse experimental data, they will often resort to a non-parametric approach. The purpose of this paper is to provide insight into a simple approach to take when it is unclear as to the appropriate parametric model and plan to conduct a Bayesian analysis. I introduce an approximate, or substitution likelihood, first proposed by Harold Jeffreys in 1939 and show how to implement the approach combined with both a non-informative and an informative prior to provide a random sample from the posterior distribution of the median of the unknown distribution. The first example I use to demonstrate the approach is a within-patient bioequivalence design and then show how to extend the approach to a parallel group design.