Prior and posterior distributions

Random variables have a prior distribution and a posterior distribution. Denote the prior by b_i (for ``before") and posterior by a<sub>i</sub> (for ``after"). Define p_i=a_i/b_i, and insert p_i in any of the inequalities above. Now suppose we have an adjustable model parameter upon which the a_i all depend. Suppose we adjust that model parameter to try to make some Jensen inequality into an equality. Thus we will be adjusting it to get all the p_i equal to each other, that is, to make all the posteriors equal to their priors. It is nice to have so many ways to do this, one for each Jensen inequality. The next question is, which Jensen inequality should we use? I cannot answer this directly, but we can learn more about the various inequalities.