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
[17]
we have
where
is the probability distribution of data
. Then from Equation (24) 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
.