Alternatively, we can apply Bayesian updating to μt, yielding
μt+ι (E) =
μt(E ∩ Et+i)
μt (Et+i)
Now we observe that, since E ∩ Et+i Ç Et+i Ç Et,
μt (Et+i)
μt(E ∩ Et+i)
μ (Et+i)
μ (Et)
μ (E ∩ Et+i)
μ (Et)
and hence,
μt+i (E)
μμ(e∩Et+i)ʌ , μμ‰η
∖ μ (Et) Γ∖ μ (Et)J
= μt+i (E) ■
6 Restricted Bayesianism
Given full rationality on a bounded domain, it is natural to consider μlt, the
restriction of the probability measure μt to events E (p) where p E Nг. That
is,
μt(E (p)) = μt(E (p))
if and only if p E Nt.
We now have two potential ways of deriving μlt+i, given the observation
of Et+i. We can use the restriction procedure at time t + 1 instead of t,
obtaining μt+i as the restriction of μt+i to Pt. Alternatively, we can apply
Bayesian updating directly to μt using the information obtained from Et+i.
The first approach yields, for any p E Pt,
μlt+i(E (p)) = μt+i(E (p))
= μ(E (p) ∩ e
μt (Et+i)
= μ (E (p) ∩ Et+i)
μ (Et+i)
where, as shown in the previous section, the last step works because E (p) ∩
Et+i Ç Et+i Ç Etso μ (Et+i) = μt (Et+i) μ (Et)