The second approach yields
tf ., P (e(p) ∩b∙+1)
p⅛ (e (p)) = — (F, )—
pt fEt+iJ
= P 4 (ri ∩ Ei+i)
~ P (Ei+1)
where the second step follows from the definition of pt. We say that restricted
Bayesian updating is consistent if, for all i,t,p
Pt+1 (E (P)) = Pt+i (E (P)) .
Suppose that, for all p,p' such that p ∈ Pt ,p' ∈ P'
p (E (p Λ p')) = p (E (p)) p (E (p')).
That is, the probabilities of propositions in the restricted domain for i are
independent of the probabilities of unconsidered propositions. It seems rea-
sonable to suppose that restricted Bayesian updating will be consistent under
these conditions. We now show that this is the case.
Proposition 1 Restricted Bayesian updating is consistent if and only if for
allp,p' such that p ∈ Pt,p' ∈ P ∖ and all possible histories hp (E (p Λ p')) =
P (E (P)) P (E (p')) .
Proof: Suppose the condition holds. Then, for all t,
P (E (pt)) = P (e (pt)) P (e (pΓ)) .
In particular,
P (E (pt+i)) = P (e (pt+ι)) P (e ((pΓ+i))) ,
and, for p ∈ Pt,
P (E (p) ∩ Et+i) = p (e (p) ∩ e (pt+ι)) P (e ((pΓ+i))) ,
so,
Pt+i (p)
p (E (p) ∩ Et+i)
p (Et+i)
P (e (p) ∩ e (pt+ι)) p (e ((pt~+tι)))
p (e (pt+ι)) P (e ((pΓ+i)))
p (e (p) ∩ e (pt+i))
P (e (pt+i))
p (e (p) ∩ Et+i)
P (Et+i)
Pt+i (p).