Motivating example: a recently completed clinical trial on migraine pains in the Johnson and Johnson pharmaceutical company
A Bi-variate Bayesian Approach
Although the contour plot of the posterior distribution sits between those of the prior distribution and the likelihood function, its projected peak is more extreme than the other two. Further examination suggests that this phenomenon is genuine in binomial clinical trials and it would not go away even if we adopt other (skewed) priors (for example, the independent beta priors used in Joseph et al. (1997)). In fact, as long as the center of a posterior distribution is not on the line joining the two centers of the joint prior and likelihood function (as it is often the case with skewed distributions), there exists a direction along which the marginal posterior fails to fall between the prior and likelihood function of the same parameter. It would be interesting to know what ramifications this counter-intuitive (or paradoxical) phenomenon may have in inferences. In any case, it is certainly not easy to explain this phenomenon to clinicians or general practitioners of statistics.