Page 207 - Introduction to Statistical Pattern Recognition

5 Parameter Estimation 189

If two distributions are normal, the Bhattacharyya distance gives an
upper bound of the Bayes error, E, as discussed in Chapter 3. The first and
second terms of (5.22), pi and p2, measure the difference between the two dis-
tributions due to the mean and covariance shifts respectively.
When Mi and i, of (5.8) and (5.9) are used to compute p, the resulting
i differs from its true value. The bias and variance of i be obtained using
can
(5.18) and (5.19).

First term pI : From (5.22), the derivatives of pI with respect to M,. are

(5.23)

(5.24)

where = (C, + C2)/2. The derivatives of pl with respect to e!;) can be
obtained from (A.31) and (A.32). That is,

where z, = (e!;) + cf))/2, x, = (h!” + h12’)/2, and m, = rn!” - m!’).

Substituting (5.23) through (5.26) into (5.18) and (5.19), and noting that
hjl) = 1 and hj2’ =A,,

(5.28)