# Maximum a posteriori

In statistics, the method of **maximum a posteriori** (MAP, posterior mode, or **maximum posterior probability**) estimation can be used to obtain a point estimate of an unobserved quantity on the basis of empirical data. It is closely related to Ronald Fisher's method of maximum likelihood (ML), but employs an augmented optimization objective which incorporates a prior distribution over the quantity one wants to estimate. MAP estimation can therefore be seen as a regularization of ML estimation.

MAP estimates can be computed in several ways:

- Analytically, when the mode(s) of the posterior distribution can be given in closed form. This is the case when conjugate priors are used.
- Via numerical optimization such as the conjugate gradient method or Newton's method. This usually requires first or second derivatives, which have to be evaluated analytically or numerically.
- Via a modification of an expectation-maximization algorithm. This does not require derivatives of the posterior density.

While MAP estimation shares the use of a prior distribution with Bayesian statistics, it is not generally seen as a Bayesian method. This is because MAP estimates are point estimates, whereas Bayesian methods are characterized by the use of distributions to summarize data and draw inferences. Bayesian methods tend to report the posterior mean or median together with posterior intervals, rather than the posterior mode. This is especially so when the posterior distribution does not have a simple analytic form: in this case, the posterior distribution can be simulated using Markov chain Monte Carlo techniques, while optimization to find its mode(s) may be difficult or impossible.

## References

- M. DeGroot,
*Optimal Statistical Decisions*, McGraw-Hill, (1970).