# Maximum A-Priori (MAP) estimation

## Maximum A-Priori Estimation

### Maximum A-Priori (MAP) estimation

Mode estimate

$$Arg max_\theta p(\theta | X)$$

Using Bayes theorem:

$$P(\theta | X)= \dfrac{P(X|\theta )P(\theta)}{P(X)}$$

So:

$$P(\theta | X)= \dfrac{P(X|\theta )P(\theta)}{P(X)}$$

$$Argmax_\theta p(\theta | X)=Argmax_\theta \dfrac{p(X|\theta )P(\theta)}{P(X)}$$

The denominator isn’t affected so:

$$Arg max_\theta p(\theta | X)=Arg max_\theta p(X|\theta )P(\theta)$$

If $$P(\theta )$$ is a constant then this is the same as the MLE estimator.

#### Other

$$Argmax_\theta p(\theta|X)$$

Mode estimate

$$p(\theta|X)= \dfrac{p(X| \theta)p(\theta )}{p(X)}$$

$$Argmax_\theta \dfrac{p(X| \theta)p(\theta )}{p(X)}$$

$$\theta$$ doesn’t change denominator so can instead use:

$$Argmax_\theta p(X| \theta)p(\theta )$$

It is the same as maximum likelihood estimator if $$p(\theta )$$ is a constant.