The name is absent

On the other hand, the condition can be false only if there exist p, h, t
such that

^At+1 ^(p) = At+1 ^(p)

that is,

At (E(p) ∩ ⅞,) , a(E (p) ∩ ⅞,)

At (E₊₁)    ^f    A (E+1)

or

At (^{e (}p⁾ ∩ ^e (pt+ι) ) = A (^{e (}p⁾ ∩ E {pt+1))

At (^e (pt+1 ))             A (^e (pt+ι))

Now since

pt+1 = pt+1 ^λ pt+1

we have

At (^e (p ^λpⁱt+ι ^λpt+ι)) = A (^e ((p ^λpt+1 ^λ pt+1)))

At (^e (pt+1 ))                A (^e (pt+1))

which implies

A (^e ((p Λ pt+1) Λ p_t+∖)) = A (^e ((p ^λ pt+1 ))) A (^e (pΓ+1))

6.1 Example

Consider a state space of the form Ω = S^,₁ × ⅛ with a product measure
A = AiA₂- Suppose that variables of potential interest are x measurable with
respect to S^,₁, + measurable with respect to S2 and

y = ;x + +,

where β is a (possibly unknown) parameter. Under the product measure
assumption, which implies independence of x and +, restricted Bayesian up-
dating is consistent for either S^,₁ or ⅛ and therefore for any propositions
about the values of x or about the values of +∙ Conversely, if x and + are
not independent, restricted Bayesian updating will not apply. We consider
the case when a set of propositions about x, sufficient to fully characterize
the distribution of x, is considered,⁴ and + represents unconsidered possible
events.

Now, what about y? We have, in the absence of any hypotheses about +,

y = a + ⅛ + ε,

⁴For example, if it is known that x is normally distributed, a set of propositions about
the mean and variance of x is required.

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