Shouldn't that be "the probability of seeing the data, _given_ that the hypothesis is correct" and "the probability of the hypothesis being correct, _given_ that the data has been observed"?
The claim is that it is reasonable to assume P(A) ~ P(B), so that P(A|B) ~ P(B|A). I am referring to P(A) and P(B) in the equation.
I agree there are debates surrounding priors, especially when you get into higher order, meta or purely hypothetical territory (there is no actual noise generator here) like this, but I think these are unavoidable.
For this to be true, P(observations) ~ P(pure noise generator) as you said earlier. Anyway, you can look at it from either perspective, doesn't really matter.
On what basis do you claim this is reasonable to expect? I suspect you have none whatsoever... in fact simple thought experiments will lead us to the opposite conclusion, and there is plenty of data telling us the same.
Ok, but I fail to see the relevance of this to anything.
>"Also P(A|B) = P(B|A) if the prior probabilities are equal, often a reasonable prior."
Here, A = Hypothesis and B = data (or vice versa). You are claiming that the "probability of seeing your data whether or not the hypothesis is correct", is often near the "probability the hypothesis is correct independent of your data"?
Based on what? What principle leads you to think these two probabilities should be near each other? BTW, if you actually have one this is like the holy grail of statistics.