Page 102 - Introduction to Statistical Pattern Recognition

84 Introduction to Statistical Pattern Recognition

Example 9: Two distributions are known to be normal, with fixed
covariance matrices XI and X2 for given expected vectors MI and M2. The
expected vectors M, and M2 are also known to be normally distributed, with
the expected vectors M and M20 and covariance matrices KI and K2. Then
according to (3.90),

1
- -(MI - M;JKT1(M; - M;o)]dM; . (3.92)
2
This can be calculated by diagonalizing X; and K; simultaneously. The result
is

1
P;(X) =
I
(2~)"'~ Xj+Kj I

I
Knowing that p;(X) is normal when p (X I Mi) and p (Mi 0;) are normal,
we can simply calculate the expected vector and covariance matrix of X assum-
ing o;:

= jM;p (M; Io;)dM;
=M;o, (3.94)

E { (X - M;o)(X - Mj0)T Io;
}
= JIJ(x-M;o)(x-M;o)~p (X 1M;)dXlp (M; Io;) dM;
(M;
= jp; + (M;-M;o)(M;-M;o)7]p lo;) dM;
=Zi+K;. (3.95)

The result is the same as (3.93).