I have a very basic question about notation—what tells me that H in the equation refers to the true hypothesis?
H stands for hypothesis. We’re taking expectations over our distribution over hypotheses: that is, expectations over which hypothesis is true.
Put another way, I don’t really understand why that equation has a different interpretation than the conservation-of-expected-evidence equation: E[P(H=hi|D)]=P(H=hi).
In the PPI inequality, the expectations are being taken over H and D jointly, in the CEE equation, the expectation is just being taken over D.
I should note that when I first saw the PPI inequality, I also didn’t get what it was saying, just because I had very low prior probability mass on it saying the thing it actually says. (I can’t quite pin down what generalisation or principle led to this situation, but there you go.)
H stands for hypothesis. We’re taking expectations over our distribution over hypotheses: that is, expectations over which hypothesis is true.
In the PPI inequality, the expectations are being taken over H and D jointly, in the CEE equation, the expectation is just being taken over D.
I should note that when I first saw the PPI inequality, I also didn’t get what it was saying, just because I had very low prior probability mass on it saying the thing it actually says. (I can’t quite pin down what generalisation or principle led to this situation, but there you go.)