The name is absent

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⁾.

More intriguing information

1. Education Responses to Climate Change and Quality: Two Parts of the Same Agenda?
2. Synthesis and biological activity of α-galactosyl ceramide KRN7000 and galactosyl (α1→2) galactosyl ceramide
3. The name is absent
4. Analyzing the Agricultural Trade Impacts of the Canada-Chile Free Trade Agreement
5. Industrial Cores and Peripheries in Brazil
6. The Shepherd Sinfonia
7. The Evolution
8. The name is absent
9. The Impact of EU Accession in Romania: An Analysis of Regional Development Policy Effects by a Multiregional I-O Model
10. Luce Irigaray and divine matter