Eliezer and Peter,
I think the problem is statics versus dynamics. Your set of equations are correct only at a specific point in time, which makes them irrelevant to saying anything about what happens later when new information arrives. That would entail subscripting H by time. For any given t, sure. But, that says nothing about what happens when new information arrives. P(H) might change.
The obvious example is indeed the black swan story, which we all know is what is lying behind this discussion. So, at a point in time before black swans are observed, let H be “all swans are white.” Perhaps there were a few folks who thought this might not be true, so say P(H) was 95%. Sure, your equations hold at a point in time, but so what? The minute the word comes in about the observation of a black swan (assuming it is accepted), P(H) just went to zero, or not much above zero, perhaps after having gradually drifted over time to 95%. Remember, your story was one about new information coming in and changes over time. But that is not what your equations are about.
This is the fatal flaw in your nice new law, Eliezer.
Eliezer and Peter, I think the problem is statics versus dynamics. Your set of equations are correct only at a specific point in time, which makes them irrelevant to saying anything about what happens later when new information arrives. That would entail subscripting H by time. For any given t, sure. But, that says nothing about what happens when new information arrives. P(H) might change.
The obvious example is indeed the black swan story, which we all know is what is lying behind this discussion. So, at a point in time before black swans are observed, let H be “all swans are white.” Perhaps there were a few folks who thought this might not be true, so say P(H) was 95%. Sure, your equations hold at a point in time, but so what? The minute the word comes in about the observation of a black swan (assuming it is accepted), P(H) just went to zero, or not much above zero, perhaps after having gradually drifted over time to 95%. Remember, your story was one about new information coming in and changes over time. But that is not what your equations are about.
This is the fatal flaw in your nice new law, Eliezer.