Here’s an example which doesn’t bear on Conservation of Expected Evidence as math, but does bear on the 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.”
taken at face value.
It’s called the Cable Guy Paradox; it was created by Alan Hájek, a philosopher the Australian National University. (I personally think the term Paradox is a little strong for this scenario.)
Here it is: the cable guy is coming tomorrow, but cannot say exactly when. He may arrive any time between 8 am and 4 pm. You and a friend agree that the probability density for his arrival should be uniform over that interval. Your friend challenges you to a bet: even money for the event that the cable guy arrives before noon. You get to pick which side of the bet you want to take—by expected utility, you should be indifferent. Here’s the curious thing: if you pick the morning bet, then almost surely there will be times in the morning when you would prefer to switch to the afternoon bet.
This would seem to be a situation in which “you can legitimately expect your confidence in a fixed proposition to be higher (on average) than before,” even though the equation P(H) = P(H|E)P(E) + P(H|~E)P(~E) is not violated. I’m not sure, but I think it’s due to multiple possible interpretations of the word “before”.
Here’s the curious thing: if you pick the morning bet, then almost surely there will be times in the morning when you would prefer to switch to the afternoon bet.
You either have a new interval, or new information suggesting the probability density for the interval has changed.
Conservation of Expected Evidence does not mean Ignorance of Observed Evidence.
This is just a restatement of the black swan problem, and it’s a non-issue. If evidence does not exist yet it does not exist yet. It doesn’t cast doubt on your methods of reasoning, nor does it allow you make a baseless guess of what might come in the future.
If you count the amount of “wanting to switch” you expect to have because the cable guy hasn’t arrived yet, it should equal exactly the amount of “wishing you hadn’t been wrong” you expect to have if you pick the second half because the cable guy arrived before your window started.
I’m not sure how to say this so it’s more easily parseable, but this equality is exactly what conservation of expected evidence describes.
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.
Here’s an example which doesn’t bear on Conservation of Expected Evidence as math, but does bear on the 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.”
taken at face value.
It’s called the Cable Guy Paradox; it was created by Alan Hájek, a philosopher the Australian National University. (I personally think the term Paradox is a little strong for this scenario.)
Here it is: the cable guy is coming tomorrow, but cannot say exactly when. He may arrive any time between 8 am and 4 pm. You and a friend agree that the probability density for his arrival should be uniform over that interval. Your friend challenges you to a bet: even money for the event that the cable guy arrives before noon. You get to pick which side of the bet you want to take—by expected utility, you should be indifferent. Here’s the curious thing: if you pick the morning bet, then almost surely there will be times in the morning when you would prefer to switch to the afternoon bet.
This would seem to be a situation in which “you can legitimately expect your confidence in a fixed proposition to be higher (on average) than before,” even though the equation P(H) = P(H|E)P(E) + P(H|~E)P(~E) is not violated. I’m not sure, but I think it’s due to multiple possible interpretations of the word “before”.
You either have a new interval, or new information suggesting the probability density for the interval has changed.
Conservation of Expected Evidence does not mean Ignorance of Observed Evidence.
This is just a restatement of the black swan problem, and it’s a non-issue. If evidence does not exist yet it does not exist yet. It doesn’t cast doubt on your methods of reasoning, nor does it allow you make a baseless guess of what might come in the future.
If you count the amount of “wanting to switch” you expect to have because the cable guy hasn’t arrived yet, it should equal exactly the amount of “wishing you hadn’t been wrong” you expect to have if you pick the second half because the cable guy arrived before your window started.
I’m not sure how to say this so it’s more easily parseable, but this equality is exactly what conservation of expected evidence describes.
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.