= A -- A N 1+ A"N -2 A-
+ _-+(--1)N'-IAm+ (-I)N'A
= (-1)" -"A
U
and
(AN'-AN')- (AN'- ._I -A )+(AN'-2- A -2)-(AN._3- A._3) Eq. 9.4
+..- + (-1)N'- (A, -A)+(-l1)N' (Ao-Ao)
= (-1)N (A, A).
U
Further, the Au's can be expressed in terms of the principal minors of I PI because those
principal minors of I P, which include row m (the 0T row) have value zero while those
which exclude row m are identical to the corresponding principal minors of I P1. In partic-
ular, the A"'s corresponding to Eq. 9.2 are
IP, =0 u=N'
A = trace(P,) =trace(P)- P(m I m) u = 1 Eq. 9.5
1 u=0
From Eq. 4.11 and Eq. 4.24, it follows that
Vi E S, Vn S' :P(i I ln)l., (1 a Pj I n) x a"('
(1+( ) jeS
-L Eq. 9.6
2
1
N
The conditional probabilities are uniform at a= 1. Then, by substituting from Eq. 9.6 into
Eq. 4.15 or 4.25, it follows that