4.2 Maximum a posteriori probability estimation
Suppose that in a given estimation problem we are not able to assign a particular cost function
. Then a natural choice is a uniform cost function equal to over a certain interval of the
parameter . From Bayes theorem  we have
where is the probability distribution of data . Then from Equation (26) one can deduce that for
each data the Bayes estimate is any value of that maximizes the conditional probability .
The density is also called the a posteriori probability density of parameter and the
estimator that maximizes is called the maximum a posteriori (MAP) estimator. It is
denoted by . We find that the MAP estimators are solutions of the following equation
which is called the MAP equation.