In Bayesian statistics, the maximum a posteriori (MAP) estimate of an unknown quantity is the mode of the posterior density. The MAP can be used to obtain a point estimate of an unobserved quantity on the basis of empirical data. It is closely related to the method of maximum likelihood (ML) estimation, but employs an augmented optimization objective which incorporates a prior density over the quantity one wants to estimate. MAP estimation is therefore a regularization of maximum likelihood estimation.

Description

be the sampling distribution. Then the function:

is known as the likelihood function and the estimate:

.

Now assume that a prior distribution as a random variable as in Bayesian statistics. We can calculate the posterior density using Bayes' theorem:

.

as the mode of the posterior density of this random variable:

The denominator of the posterior density (the marginal likelihood is a constant function), which occurs whenever the prior distribution is taken as the reference measure, as is typical in function-space applications.

In the context of Bayes estimators, the MAP can be recovered as the minimizer of the Bayes risk with risk function

is quasi-concave. is discrete.

Computation

MAP estimates can be computed in several ways:

1. Analytically, when the mode(s) of the posterior density can be given in closed form. This is the case when conjugate priors are used.

2. 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.

3. Via a modification of an expectation-maximization algorithm. This does not require derivatives of the posterior density.

4. Via a Monte Carlo method using simulated annealing

Limitations

While only mild conditions are required for MAP estimation to be a limiting case of Bayes estimation (under the 0–1 loss function), it is not representative of Bayesian methods in general. This is because MAP estimates are point estimates, and depend on the arbitrary choice of reference measure, whereas Bayesian methods are characterized by the use of distributions to summarize data and draw inferences: thus, Bayesian methods tend to report the posterior mean or median instead, together with credible intervals. This is both because these estimators are optimal under squared-error and linear-error loss respectively—which are more representative of typical loss functions—and for a continuous posterior distribution there is no loss function which suggests the MAP is the optimal point estimator.

In many types of models, such as mixture models, the posterior may be multi-modal. In such a case, the usual recommendation is that one should choose the highest mode: this is not always feasible (global optimization is a difficult problem), nor in some cases even possible (such as when identifiability issues arise). Furthermore, the highest mode may be uncharacteristic of the majority of the posterior, especially in many dimensions.

Finally, unlike ML estimators, the MAP estimate is not invariant under reparameterization. Switching from one parameterization to another involves introducing a Jacobian that impacts on the location of the maximum. In contrast, Bayesian posterior expectations are invariant under reparameterization.

as negative.

Example

of IID random variables. Note that the normal distribution is its own conjugate prior, so we will be able to find a closed-form solution analytically.

The function to be maximized is then given by

:

Thus, we see that the MAP estimator for μ is given by

which turns out to be a linear interpolation between the prior mean and the sample mean weighted by their respective covariances.

is called a non-informative prior and leads to an improper probability distribution

References