Page 62 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

BAYESIAN ESTIMATION 51

. uniform cost function:

1 if ^ x xk >
x
k
x
Cð^ xjxÞ¼ 1 with: ! 0 ð3:8Þ
x
0 if ^ x xk
k
1
The first two cost functions are instances of the Minkowski distance
measures (see Appendix A.2). The third cost function is an approxima-
tion of the distance measure mentioned in (a.22).
Risk minimization
With an arbitrarily selected estimate ^ x and a given measurement vector
x
z, the conditional risk of ^ x is defined as the expectation of the cost
x
function:
Z
x
x
x
Rð^ xjzÞ¼ E½Cð^ xjxÞjzŠ¼ Cð^ xjxÞpðxjzÞdx ð3:9Þ
x
In Bayes estimation (or minimum risk estimation) the estimate is the
parameter vector that minimizes the risk:

f
^ x xðzÞ¼ argmin RðxjzÞg ð3:10Þ
x
The minimization extends over the entire parameter space.
x
The overall risk (also called average risk) of an estimator ^ x(z) is the
expected cost seen over the full set of possible measurements:

Z
x
x
R ¼ E½Rð^ xðzÞjzފ ¼ Rð^ xðzÞjzÞpðzÞdz ð3:11Þ
z
Minimization of the integral is accomplished by minimization of the
integrand. However, since p(z) is positive, it suffices to minimize
x
R(^ x(z)jz). Therefore, the Bayes estimator not only minimizes the condi-
tional risk, but also the overall risk.
The Bayes solution is obtained by substituting the selected cost func-
tion in (3.9) and (3.10). Differentiating and equating it to zero yields for
the three cost functions given in (3.6), (3.7) and (3.8):

. MMSE estimation (MMSE ¼ minimum mean square error).
. MMAE estimation (MMAE ¼ minimum mean absolute error).
. MAP estimation (MAP ¼ maximum a posterior).