Page 66 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

BAYESIAN ESTIMATION 55

The conditional risk of this solution is the sum of the variances of the
estimated parameters:

Z
T
x
x
x
Rð^ x MMSE ðzÞjzÞ¼ ð^ x MMSE ðzÞ xÞ ð^ x MMSE ðzÞ xÞpðxjzÞdx
x
Z
T
¼ ðE½xjzŠ xÞ ðE½xjzŠ xÞpðxjzÞdx ð3:14Þ
x
M 1
X
¼ Var½x m jzŠ
m¼0
Hence the name ‘minimum variance estimator’.

3.1.2 MAP estimation

If the uniform cost function is chosen, the conditional risk (3.9)
becomes:

Z
x
x
Rð^ xjzÞ¼ pðxjzÞdx pð^ xjzÞ
x ð3:15Þ
x
¼ 1 pð^ xjzÞ
The estimate which now minimizes the risk is called the maximum a
posterior (MAP) estimate:

^ x x MAP ðzÞ¼ argmaxfpðxjzÞg ð3:16Þ
x

This solution equals the mode (maximum) of the posterior probability. It
can also be written entirely in terms of the prior probability densities and
the conditional probabilities:

pðzjxÞpðxÞ
^ x x MAP ðzÞ¼ argmax ¼ argmaxfpðzjxÞpðxÞg ð3:17Þ
x pðzÞ x

This expression is similar to the one of MAP classification; see (2.12).