Is maximum a posteriori estimation some kind of Bayesian Estimation?
If yes, can you point out other Bayesian estimators?
Edit:
So I've come to know the following (don't know if they are correct):
MAP finds the mode of the posterior, and Bayesian Estimator finds the distribution of that posterior with respect to the estimating parameters. And in order for Bayesian Estimator to find the optimal parameters, some particular risk function (like Mean Squared Error, 0/1 error etc.) must be defined.
So it seems these two estimators have nothing in common except they both incorporate prior into their estimation. Or is it possible that MAP is some special case of Bayesian Estimator with some particular risk function.
Best Answer
MAP finds the mode of the posterior, while full Bayes characterizes the entire distribution -- the probability assigned to each element, whence we can know all of its moments and so on. Chapter 4 of Gelman's Bayesian Data Analysis 3rd ed develops this point in more detail.