MAP Decoder

**MAP Decoder** is defined, in Bayesian statistics, that it is a decoder whose estimate is the argument of the maximization problem
of the A posteriori function. The MAP problem can be written as follows: $$\hat{\theta}_{MAP}=arg\;\underset{\theta}{max}\;f(\theta|x)$$

Using the Bayesian formula: $$f(\theta|x)=\dfrac{f(x|\theta)\cdot f(\theta)}{f(x)}$$,

The **MAP** estimate can be written as: $$\hat{\theta}_{MAP}=arg\;\underset{\theta}{max}\;f(x|\theta)\cdot f(\theta)$$

where $f(\theta|x)$ is the A posteriori function, $f(x|\theta)$ is the likelihood function and $f(\theta)$ represents the prior information about $\theta$. Obviously, the **MAP** decoder simplifies to **Maximum Likelihood (ML)** decoder in case of constant prior information.

You can find more details about **MAP**- and **ML** decoder in [3].