Let (O,A) where O=state space and A = algebra on the state space.
A"=>"B => A more likely than B.
p() is a function that represents "=>":
A"=>"B <=> p(A)=>p(B)
Properties of p:
* normalized => p(empty set)=0 p(O)=1
* additive A intersect B = empty set => p(A union B)=p(A)+p(B)
P is unique <=> there exists n-1 Ai ~ Bi in A linearly independent.
Theorem:
A an algebra on O, "=>" a relation on A =>
* there exists P:O->R which represents "=>" <=> "=>" satisfies QP1, QP4
* "=>" satisfies QP1,QP2,QP3 /\ P(empty set)=0 /\ P(O)=1 => forall B in A: 1=>P(B)=>0
* "=>" satisfies QP1,QP2,QP3,QP4 => there exists P /\ there exists G:R^2->R s.t. C intersect B = empty set, P(A union B)=G(P(A),P(B))
* G(X,Y)=G(Y,X),
* G strictly increasing in x, G(x,p(empty set))=x,
* G associative G(G(x,y),z)=G(x,G(y,z))
The function G sometimes reduces to addition.
Theorem: O finite, A in 2^O =>
* there exists p:O->R representing "=>" and additive <=> "=> complete and QP5
* "=>" satisfies QP2 also => P(empty set) = 0 /\ P(O)=1 /\ pi=P(wi)=>0, Sum(pi)=1