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It might seem that minimization of is equivalent to minimization of . If these two minimization problems were equivalent, then maximization of would be the best method in statistical estimation. However, minimization and expectation cannot be commutative.
Hence maximization of does not mean minimization of . This is the basic reason why statistical learning does not result in a simple optimization problem.
In statistical estimation, the pair is statistical model which is optimized for given random samples. Hence, if is fixed and is made coordinate-free, such a pair is not appropriate for statistical estimation in general.
Although the MAP employs an a priori distribution, its generalization error is quite different from that of Bayes estimation.