# 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.

### MAP of the Gaussian distribution