A: first theory
B: second theory
D: data accumulated between the 10th and 20th trials
We're interested in the ratio P(A|D)/P(B|D).
By Bayes' Theorem:
P(A|D) = P(D|A)P(A)/P(D)
P(B|D) = P(D|B)P(B)/P(D)
Then
P(A|D)/P(B|D) = P(D|A)P(A)/(P(D|B)P(B)).
If each theory predicts the data observed with equal likelihood (that is, under neither theory is the data more likely to be observed), then P(D|A) = P(D|B) so we can simplify,
P(A|D)/P(B|D) = P(A)/P(B) >> 1
given that presumably theory A was a much more plausible pri... (read more)
My approach: (using Bayes' Theorem explicitly)
A: first theory
B: second theory
D: data accumulated between the 10th and 20th trials
We're interested in the ratio P(A|D)/P(B|D).
By Bayes' Theorem:
P(A|D) = P(D|A)P(A)/P(D)
P(B|D) = P(D|B)P(B)/P(D)
Then
P(A|D)/P(B|D) = P(D|A)P(A)/(P(D|B)P(B)).
If each theory predicts the data observed with equal likelihood (that is, under neither theory is the data more likely to be observed), then P(D|A) = P(D|B) so we can simplify,
P(A|D)/P(B|D) = P(A)/P(B) >> 1
given that presumably theory A was a much more plausible pri... (read more)