Example
Suppose that we are given a sequence of IID random variables and an a priori distribution of is given by . We wish to find the MAP estimate of .
The function to be maximized is then given by
which is equivalent to minimizing the following function of :
Thus, we see that the MAP estimator for μ is given by
which turns out to be a linear interpolation between the prior mean and the sample mean weighted by their respective covariances.
The case of is called a non-informative prior and leads to an ill-defined a priori probability distribution; in this case
Read more about this topic: Maximum A Posteriori Estimation
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