Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

9.15 MAXIMUM A POSTERIORI ESTIMATION

Suppose we want to estimate random parameter Θ from N iid samples assuming that it has known pdf . Recall that in the context of estimation, is called the prior distribution which characterizes Θ before any samples are observed. The posterior distribution characterizes Θ after collecting N samples. Using Bayes' formula, the posterior distribution is expressed in terms of the prior distribution as follows:

(9.240)

The numerator is the joint pdf of the samples given that Θ = θ, whereas the denominator is the joint pdf of the samples without any conditioning. Generally, the two terms in the numerator are assumed to be known. The denominator, on the other hand, is not known, but can be derived from the numerator as follows:

(9.241)

The maximum a posteriori (MAP) estimator is the value maximizing the posterior distribution on the left-hand side of ( ...