It is well known that the classical Bayesian posterior arises naturally as the unique solution of different optimization
problems, without the necessity of interpreting data as conditional probabilities and then using Bayes\u2019 Theorem. Here it
is shown that the Bayesian posterior is also the unique minimax optimizer of the loss of self-information in combining the
prior and the likelihood distributions, and is the unique proportional consolidation of the same distributions. These results,
direct corollaries of recent results about conflations of probability distributions, further reinforce the use of Bayesian
posteriors, and may help partially reconcile some of the differences between classical and Bayesian statistics