What's the difference between Maximum a posteriori and Bayes' rule? They look similar, except that I do understand Bayes' rule and I don't understand MAP. The people I asked - who work in math and computer science - have never heard about Maximum a posteriori.
MAP estimator (or maximum a posteriori) is indeed Bayesian estimator. It minimizes Bayes risk for a "hit-or-miss" cost function. MAP is very close to MLE (indeed if the amount of samples goes to infinity then MAP and MLE equivalent). MAP is generally used because in some cases it allows to perform numerically efficient estimations, comparing to MLE or others. For more details you can turn to some textbooks (Kay S.M. "Fundamentals of statistical signal processing estimation theory", for example). It is said there (Ch. 11.3), that minimization of the Bayes risk with quadratic error/"hit-or-miss" error (cost function) turns to MMSE/MAP.