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:

$\hat{x}^{MAP} = \arg\max \ (p_{X \mid Z_{k}}(x \mid z_{k}))$ $= \arg\max \ (p_{Z_{k}\mid X}(z_{k} \mid x)p_{X}(x))$