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 pt+∖)) = 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.