MAP vs Bayes Estimator When it comes to Bayesian Inference, we are interested in estimating the Posterior Distribution of some model parameter(s) based on the data we observe and some Prior Probability Distribution.
After defining the form of the Posterior Distribution, we are usually interested in one of two things:

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*MAP Estimator (https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation): This is the "mode" of the Posterior Distribution (i.e. most common point)

*Bayes Estimator (https://en.wikipedia.org/wiki/Bayes_estimator): This is the "expected value" of the Posterior Distribution (note: I think this is closer to what is typically done in Frequentist Estimation)

My Question: I am trying to understand in what kinds of situations would we be more interested in the MAP Estimator and in what kinds of situations would be more interested in the Bayes Estimator.
Informally looking at references, it seems like the MAP Estimator tends to be used more often than the Bayes Estimator - but I can not seem to find a (simpler) reference or understand why this is the case.
For example - suppose I flip a coin 100 times and observe "n" heads. I happen to believe that some specific Beta Distribution (e.g. Beta ~ a,b) is well suited to be a prior. Thus, I can now define the Posterior Distribution for "p" (the probability of success). As I understand, I have the option of finding out the determining the MAP of "p" or the Bayes Estimator of "p". What are the advantages and the disadvantages of choosing either of these estimators for "p"?
Can someone please help me understand this?
Thanks!

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*Note: Suppose I have 10 numbers that represent the number of lakes in countries on some continent : 234, 132, 87, 76, 199, 501, 612, 882, 0,0 .

*The "mode" of these numbers is 0 and the "mean" is 272.

*I could  say that the average number of lakes for countries in this continent is 272 whereas 20% of countries in this continent have no lakes.

*At first, I would have argued that in this case, the "mean" (analogous to the concept of the Bayes Estimator) provides us with "more useful information" when compared to the "mode" (analogous to the concept of the MAP) - but after thinking about this some more, I am not so sure.

*I think the choice of "MAP" vs "Bayes Estimator" might depend on the specific question of interest - but I am not sure how I can conceptualize what types of questions are better answered with "MAP" vs "Bayes Estimator"


Let's clarify a little:

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*The MAP estimate is the posterior mode. It is the point $\hat{\theta} = \underset{\theta \in \mathbb{R}^p}{\arg\max} \log(P(\theta \mid y))$.


*The Bayes estimator is not necessarily the posterior expectation.  According to the reference you linked to, the Bayes estimator is the minimizer of the posterior expected loss.  If the loss function is the mean squared error, then the posterior expected value minmizes the MSE and so hence the posterior expected value is the bayes estimator.  Change the loss and the bayes estimator might change.

My Question: I am trying to understand in what kinds of situations would we be more interested in the MAP Estimator and in what kinds of situations would be more interested in the Bayes Estimator.

I think to a first approximation...

*

*You'd be more interested in the MAP estimate if you're fitting a model and want to get an estimate of the parameters of said model.  I do something like that in this post.  That isn't to say MAP is always good for fitting models, and in fact it can be really bad when using hierarchical or high dimensional models.


*You'd be more interested in the Bayes estimator when you want your estimator to have some specific property.  As an example, maybe you want your estimate to be robust to outliers, in which case you might choose the sum of absolute deviations to minimize.
