Criticism
While MAP estimation is a limit of Bayes estimators (under the 0-1 loss function), it is not very representative of Bayesian methods in general. This is because MAP estimates are point estimates, whereas Bayesian methods are characterized by the use of distributions to summarize data and draw inferences: thus, Bayesian methods tend to report the posterior mean or median instead, together with credible intervals. This is both because these estimators are optimal under squared-error and linear-error loss respectively - which are more representative of typical loss functions - and because the posterior distribution may not have a simple analytic form: in this case, the distribution can be simulated using Markov chain Monte Carlo techniques, while optimization to find its mode(s) may be difficult or impossible.
In many types of models, such as mixture models, the posterior may be multi-modal. In such a case, the usual recommendation is that one should choose the highest mode: this is not always feasible (global optimization is a difficult problem), nor in some cases even possible (such as when identifiability issues arise). Furthermore, the highest mode may be uncharacteristic of the majority of the posterior.
Finally, unlike ML estimators, the MAP estimate is not invariant under reparameterization. Switching from one parameterization to another involves introducing a Jacobian that impacts on the location of the maximum.
As an example of the difference between Bayes estimators mentioned above (mean and median estimators) and using an MAP estimate, consider the case where there is a need to classify inputs as either positive or negative (for example, loans as risky or safe). Suppose there are just three possible hypotheses about the correct method of classification, and with posteriors 0.4, 0.3 and 0.3 respectively. Suppose given a new instance, classifies it as positive, whereas the other two classify it as negative. Using the MAP estimate for the correct classifier, is classified as positive, whereas the Bayes estimators would average over all hypotheses and classify as negative.
Read more about this topic: Maximum A Posteriori Estimation
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