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].