IJ Good's paper (1967) gives a solution to Ayer's problem on why we should make new observations.
I'm trying to follow the steps in his solution.
His assumptions are that:
There are r mutually exclusive and exhaustive hypothesis H1, H2,..., Hr.
On some evidence E the prior probabilities are pi = P(H|E).
There's an observation that has possible outcomes E1, E2,..., Et, where P(Ek| Hi) = pik (i = 1, 2,..., r; k = 1, 2, ..., t).
Then he lets qik = P(Hi|E^Ek) = pipik/ Sigmai(pipik) be the posterior of Hi if Ek occurs.
This is not the posterior I get if I use Bayes Theorem:
P(Hi|E^Ek) = P(Ek|Hi^E) P(Hi|E) / Sigmai(P(Ek|Hi^E) P(Hi|E))
is not equal to:
P(Ek|Hi) P(Hi|E) / Sigmai(P(Ek|Hi) P(Hi|E)
Which is what's implied by Good's formula.
Where's my mistake? Does P(Ek|Hi^E) = P(Ek|Hi) in this formula? Why?
Thanks for your time.