At 10am tomorrow, I can legitimately express my confidence in the proposition “the cable guy will arrive after noon” is different to what it was today.
There are two cases to consider:
The cable guy arrived before 10am (occurs with 25% probability). In this case, I expect that he has a close on zero probability of arriving after noon.
The cable guy is known not to have arrived before 10am (occurs with 75% probability). At this point, I calculate that the odds of the cable guy turning up after noon are two in three.
But none of this takes anything away from the original statement:
“There is no possible plan you can devise, no clever strategy, no cunning device, by which you can legitimately expect your confidence in a fixed proposition to be higher (on average) than before.”
This is because I am changing my probability estimate on the basis of new information received—it’s not a fixed proposition.
It may not be obvious at first glance why CCC’s response supports Eliezer’s point, so to spell it out in more detail: as of now (the night before), I expect that as of 10 am, there will be
A. 25% that cable guy already arrived, ie 25% chance of ‘before noon and before 10’.
B. 75% that cable guy has not already arrived, and 33.3% chance that he’ll arrive between 10 am and noon, ie .75 / 3 = 25% chance of ‘before noon and after 10’.
C. 75% that cable guy has not already arrived, and 66.7% chance that he’ll arrive after noon, ie .75 * 2 / 3 = 50% chance of ‘after noon’.
This illustrates Eliezer’s point nicely. Since the total ‘before noon’ chance is case A (25%) + case B (25%) and the total ‘after noon’ chance is case C (50%), my currently expected belief at 10 am tomorrow is the same as my expected belief now, and so we don’t expect our confidence in ‘before noon’ to be higher (or lower) than it is now.
As of 10 am tomorrow, we will (as CCC says) have changed our probabilities in one direction or another, but in our current expectation of our beliefs at that point, it’s still 50:50.
At 10am tomorrow, I can legitimately express my confidence in the proposition “the cable guy will arrive after noon” is different to what it was today.
There are two cases to consider:
The cable guy arrived before 10am (occurs with 25% probability). In this case, I expect that he has a close on zero probability of arriving after noon.
The cable guy is known not to have arrived before 10am (occurs with 75% probability). At this point, I calculate that the odds of the cable guy turning up after noon are two in three.
But none of this takes anything away from the original statement:
This is because I am changing my probability estimate on the basis of new information received—it’s not a fixed proposition.
It may not be obvious at first glance why CCC’s response supports Eliezer’s point, so to spell it out in more detail: as of now (the night before), I expect that as of 10 am, there will be
A. 25% that cable guy already arrived, ie 25% chance of ‘before noon and before 10’.
B. 75% that cable guy has not already arrived, and 33.3% chance that he’ll arrive between 10 am and noon, ie .75 / 3 = 25% chance of ‘before noon and after 10’.
C. 75% that cable guy has not already arrived, and 66.7% chance that he’ll arrive after noon, ie .75 * 2 / 3 = 50% chance of ‘after noon’.
This illustrates Eliezer’s point nicely. Since the total ‘before noon’ chance is case A (25%) + case B (25%) and the total ‘after noon’ chance is case C (50%), my currently expected belief at 10 am tomorrow is the same as my expected belief now, and so we don’t expect our confidence in ‘before noon’ to be higher (or lower) than it is now.
As of 10 am tomorrow, we will (as CCC says) have changed our probabilities in one direction or another, but in our current expectation of our beliefs at that point, it’s still 50:50.