MAP Estimator

MAP(Maximum A Posteriori) Estimator

According to the Bayes Theorem, a probability density function of an unknown random vector $X$ conditioned on a measurement vector $Z_k=z_k$ is given as follows:

$p_{X\mid Z_k}(x\mid z_k)=\frac{p_{Z_k\mid X}(z_k\mid x)p_X(x)}{p_{Z_k}(z_k)}$

where

• $p_X(x)$ : A probability density function of $X$ known a priori before a vector $Z_k$ is measured as $z_k$

• $p_{Z_k}(z_k)$ : A probability density function of the set of measurement vectors $Z_k$, representing the probability information of the measurement process.

• $p_{Z_k\mid X}(z_k\mid x)$ : A conditional probability density function of $Z_k$ conditioned on $X=x$. It is a likelihood function that represents how often a specific set of measurement vectors $z_k$ appears depending on $x$.

• $p_{X\mid Z_k}(x\mid z_k)$: A conditional probability density function of X given as $Z_k=z_k$ post-measurement.
  
  

MAP estimator is defined as the estimation value of $X$ when the conditional probability density function of the unknown random vector $X$ is at its maximum value. As an image:

As a formula:

${\hat{x}}^{MAP}=\arg max(p_{X\mid Z_k}(x\mid z_k))$ $=\arg max(p_{Z_k\mid X}(z_k\mid x)p_X(x))$