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The second approach yields

_tf ., P (e(p) ∩b∙₊₁)

p⅛ ⁽e (p)) = — ⁽_F, ⁾—

^pt f^Et+iJ

= P 4 (ri ∩ ^Ei+i)

~ P (Ei₊₁)

where the second step follows from the definition of p_t. 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 ∈ P^t ,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 ∈ P^t,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 ∈ P^t,

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 (p^t+ι)) p (^e ((p_t~₊^tι)))
p (^e (pt+ι)) P (^e ((pΓ+i)))

p (^e⁽p⁾ ∩ ^e (pt+i))

P (^e (pt+i))

p (^e⁽p⁾ ∩ ^E^t+i)

P (Et+i)

Pt+i ⁽p⁾.

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