  Prentice & Pyke (1979) established that the maximum likelihood estimate of an
odds-ratio in a case-control study is the same as would be found by running a
logistic regression: in other words, for this specific target the incorrect
prospective model is inferentially equivalent to the correct retrospective
model. Similar results have been obtained for other models, and conditions have
also been identified under which the corresponding Bayesian property holds,
namely that the posterior distribution of the odds-ratio be the same, whether
computed using the prospective or the retrospective likelihood. Here we
demonstrate how these results follow directly from certain parameter
independence properties of the models and priors, and identify prior laws that
support such reverse analysis, for both standard and stratified designs.
