So just to be clear. There are two things, the prior probability, which is the value P(H|I), and the back ground information which is 'I'. So P(H|D,I1) is different from P(H|D,I2) because they are updates using the same data and the same hypothesis, but with different partial background information, they are both however posterior probabilities. And the priors P(HI1) may be equal to P(H|I2) even if I1 and I_2 are radically different and produce updates in opposite directions given the same data. P(H|I) is still called the prior probability, but it is smething very differnet from the background information which is essentially just I.

Is this right? Let me be more specific.

Let's say my prior information is case1, then P( second ball is R| first ball is R & case1) = 4/9

If my prior information was case2, then P( second ball is R| first ball is R & case2) = 2/3 [by the rule of succession]

and P( first ball is R| case1) = 50% = P( first ball is R|case2)

This is why different prior information can make you learn in different directions, even if two prior informations produce the same prior probability?

Please let me know if i am making any sort of mistake. Or if I got it right, either way.