Maximum a posteriori probability estimation

4.2 Maximum a posteriori probability estimation

Suppose that in a given estimation problem we are not able to assign a particular cost function $C (θ′,θ)$ . Then a natural choice is a uniform cost function equal to $0$ over a certain interval $Iθ$ of the parameter $θ$ . From Bayes theorem [20 ] we have

where $p(x)$ is the probability distribution of data $x$ . Then from Equation (26

) one can deduce that for each data $x$ the Bayes estimate is any value of $θ$ that maximizes the conditional probability $p(θ|x)$ . The density $p(θ|x )$ is also called the a posteriori probability density of parameter $θ$ and the estimator that maximizes $p(θ|x)$ is called the maximum a posteriori (MAP) estimator. It is denoted by $ˆ θMAP$ . We find that the MAP estimators are solutions of the following equation

which is called the MAP equation.