e, and these are described in the next three sections. 10.2.3 Estimating a Given r, the First Way The first way is called the minimum mean squared error (MMSE) method. It gener- ates an estimate ammes called the minimum mean squared error estimate. The idea here is to output the estimate of a, mmse , , that is, on average, as close to a as possible. Mathematically, the idea is to output the am mmse that satisfies the following state- ment: The value amme mmse minimizes the value E[(a-amme)2] (where E[x] is the expected value of x). 282 Chapter Ten After some work, you can show that the minimum mean squared estimate amme can be calculated with a simple mathematical equation if you know the probabilities p(ra), p(a),and p (r): (10.7) where p(alp)_ 10.2.4 Estimating a Given r, the Second Way Often, the mathematical computation of the estimate mmse from Equation (10.7) rather messy. So, engineers, who have never been big on "messy math," found a new way to get an estimate of a. They called their new way the maximum a posteriori, or MAP, estimate of a, and labeled it This is the estimate that satisfies the following mathematical statement: Choose the value âMAP that minimizes E[C(a, MAP) where C(a, ) = 1 if is not very close to a, and very close to a. This mathematical statement basically says, in words, find the âMAP that is very close to a. The value of âMAP can be found if you know the two probabilities p(r)a) and p(a). If you know these two things, then the value of can be found according to the mathematical statement: is the value that maximizes the function p(r)a) p(a); or, equivalently through the mathematical equation (10.8) (10.9) Generally speaking, the estimates ammes mmse and work out to the same value. In fact, in almost every case of practical interest, these values are identical. Estimation and Synchronization 10.2.5 Estimating a Given r, the Third Way There is one more way to generate an estimate for a given r. This method is used if you have a slightly different case than that we have been studying SO