Page 108 - Introduction to Statistical Pattern Recognition

90 Introduction to Statistical Pattern Recognition

we can carry out the integration numerically, even if F,(o) is a compli-
cated function and not integrable explicitly.

Example 10: In order to confirm the above discussion, let us study a
simple case with two normal distributions: Nx(O,I) and Nx(M,I). From (3.12),
the Bayes classifier becomes
1
h (X) = MTX - -MTM , (3.1 15)
2
and the ol-density function Nx(O,I) is expressed by
1 1
Pl(X) = exp[--xTx]. (3.1 16)
(2X)"'* 2
~

1 1
= INx( joM,I)exp - -joMTM dX
2

o2
= exp I--MTM - (3.1 17)
2
Equation (2.24) shows the characteristic function of a multivariate normal dis-
tribution, and (3.117) is a special case for one-dimension. Therefore, taking
the inversion,
GdiZii [- (h + MTM22)2 (3.1 18)
1
Ph(h 101) = 2MTM

That is, the w,-distribution of h has E(hlol ) =-MTM/2 and
1
Var( h I ol = MTM, which are identical to (3.97) and (3.99) for M 1=0, &=M,
and CI=C2=I,