LINK: Infinity, probability and disagreement

Alejandro1

I saw this conundrum at Alexander Pruss's blog and I thought LWers might enjoy discussing it:

Consider the following sequence of events:

You roll a fair die and it rolls out of sight.

An angel appears to you and informs you that you are one of a countable infinity of almost identical twins who independently rolled a fair die that rolled out of sight, and that similar angels are appearing to them all and telling them all the same thing. The twins all reason by the same principles and their past lives have been practically indistinguishable.

The angel adds that infinitely many of the twins rolled six and infinitely many didn't.

The angel then tells you that the angels have worked out a list of pairs of identifiers of you and your twins (you're not exactly alike), such that each twin who rolled six is paired with a twin who didn't roll six.

The angel then informs you that each pair of paired twins will be transported into a room for themselves. And, poof!, it is so. You are sitting across from someone who looks very much like you, and you each know that you rolled six if and only if the other did not.

Let H be the event that you did not roll six. How does the probability of H evolve?

After step 1, presumably your probability of H is 5/6. But after step 5, it would be very odd if it was still 5/6. For if it is still 5/6 after step 5, then you and your twin know that exactly one of you rolled six, and each of you assigns 5/6 to the probability that it was the other person who rolled six. But you have the same evidence, and being almost identical twins, you have the same principles of judgment. So how could you disagree like this, each thinking the other was probably the one who rolled six?

Thus, it seems that after step 5, you should either assign 1/2 or assign no probability to the hypothesis that you didn't get six. And analogously for your twin.

But at which point does the change from 5/6 to 1/2-or-no-probability happen? Surely merely physically being in the same room with the person one was paired with shouldn't have made a difference once the list was prepared. So a change didn't happen in step 5.

And given 3, that such a list was prepared doesn't seem at all relevant. Infinitely many abstract pairings are possible given 3. So it doesn't seem that a change happened in step 4. (I am not sure about this supplementary argument: If it did happen after step 4, then you could imagine having preferences as to whether the angels should make such a list. For instance, suppose that you get a goodie if you rolled six. Then you should want the angels to make the list as it'll increase the probability of your having got six. But it's absurd that you increase your chances of getting the goodie through the list being made. A similar argument can be made about the preceding step: surely you have no reason to ask the angels to transport you! These supplementary arguments come from a similar argument Hud Hudson offered me in another infinite probability case.)

Maybe a change happened in step 3? But while you did gain genuine information in step 3, it was information that you already had almost certain knowledge of. By the law of large numbers, with probability 1, infinitely many of the rolls will be sixes and infinitely many won't. Simply learning something that has probability 1 shouldn't change the probability from 5/6 to 1/2-or-no-probability. Indeed, if it should make any difference, it should be an infinitesimal difference. If the change happens at step 3, Bayesian update is violated and diachronic Dutch books loom.

So it seems that the change had to happen all at once in step 2. But this has serious repercussions: it undercuts probabilistic reasoning if we live in multiverse with infinitely many near-duplicates. In particular, it shows that any scientific theory that posits such a multiverse is self-defeating, since scientific theories have a probabilistic basis.

I think the main alternative to this conclusion is to think that your probability is still 5/6 after step 5. That could have interesting repercussions for the disagreement literature.

I saw this conundrum at Alexander Pruss's blog and I thought LWers might enjoy discussing it:

Consider the following sequence of events:

You roll a fair die and it rolls out of sight.

An angel appears to you and informs you that you are one of a countable infinity of almost identical twins who independently rolled a fair die that rolled out of sight, and that similar angels are appearing to them all and telling them all the same thing. The twins all reason by the same principles and their past lives have been practically indistinguishable.

The angel adds that infinitely many of the twins rolled six and infinitely many didn't.

The angel then tells you that the angels have worked out a list of pairs of identifiers of you and your twins (you're not exactly alike), such that each twin who rolled six is paired with a twin who didn't roll six.

The angel then informs you that each pair of paired twins will be transported into a room for themselves. And, poof!, it is so. You are sitting across from someone who looks very much like you, and you each know that you rolled six if and only if the other did not.

Let H be the event that you did not roll six. How does the probability of H evolve?

After step 1, presumably your probability of H is 5/6. But after step 5, it would be very odd if it was still 5/6. For if it is still 5/6 after step 5, then you and your twin know that exactly one of you rolled six, and each of you assigns 5/6 to the probability that it was the other person who rolled six. But you have the same evidence, and being almost identical twins, you have the same principles of judgment. So how could you disagree like this, each thinking the other was probably the one who rolled six?

Thus, it seems that after step 5, you should either assign 1/2 or assign no probability to the hypothesis that you didn't get six. And analogously for your twin.

But at which point does the change from 5/6 to 1/2-or-no-probability happen? Surely merely physically being in the same room with the person one was paired with shouldn't have made a difference once the list was prepared. So a change didn't happen in step 5.

And given 3, that such a list was prepared doesn't seem at all relevant. Infinitely many abstract pairings are possible given 3. So it doesn't seem that a change happened in step 4. (I am not sure about this supplementary argument: If it did happen after step 4, then you could imagine having preferences as to whether the angels should make such a list. For instance, suppose that you get a goodie if you rolled six. Then you should want the angels to make the list as it'll increase the probability of your having got six. But it's absurd that you increase your chances of getting the goodie through the list being made. A similar argument can be made about the preceding step: surely you have no reason to ask the angels to transport you! These supplementary arguments come from a similar argument Hud Hudson offered me in another infinite probability case.)

Maybe a change happened in step 3? But while you did gain genuine information in step 3, it was information that you already had almost certain knowledge of. By the law of large numbers, with probability 1, infinitely many of the rolls will be sixes and infinitely many won't. Simply learning something that has probability 1 shouldn't change the probability from 5/6 to 1/2-or-no-probability. Indeed, if it should make any difference, it should be an infinitesimal difference. If the change happens at step 3, Bayesian update is violated and diachronic Dutch books loom.

So it seems that the change had to happen all at once in step 2. But this has serious repercussions: it undercuts probabilistic reasoning if we live in multiverse with infinitely many near-duplicates. In particular, it shows that any scientific theory that posits such a multiverse is self-defeating, since scientific theories have a probabilistic basis.

I think the main alternative to this conclusion is to think that your probability is still 5/6 after step 5. That could have interesting repercussions for the disagreement literature.

Probability theory doesn't automatically work on infinite sets. If you approach this problem as the well-defined limit of a finite problem, the answer is simple.

You roll a fair die and it rolls out of sight. You assign a probability of 1/6 to the proposition that you rolled a heads, since you have no reason to suspect any particular face came up.
An angel appears to you and informs you that you are one of N almost identical twins who independently rolled a fair die that rolled out of sight, and that similar angels are appearing to them all and telling them all the same thing. The twins all reason by the same principles and their past lives have been practically indistinguishable. Still 1/6, no new relevant information here.
The angel adds that X of the twins rolled six and N-X didn't. Woah, something is messed up, unless X is approximately N/6...
The angel then tells you that the angels have worked out a list of pairs of identifiers of you and your twins (you're not exactly alike), such that each twin who rolled six is paired with a twin who didn't roll six. Yep, something is messed up. We have X=N/2, which is nowhere near N/6. A few things could happen now, depending on our priors. In real life, we would probably stop trusting the angel, stop trusting that the dice are fair, or both; it depends on how large N is (and hence how unlikely the "even split with fair dice" is), and how strong our faith was in the angel and the dice. Not in real life, following the conceit of the problem, an incredibly unlikely event just occurred. But no matter; we confidently assign a probability of 1/2 to having rolled a six, based on what the angel told us and no other information to identify our roll.
The angel then informs you that each pair of paired twins will be transported into a room for themselves. And, poof!, it is so. You are sitting across from someone who looks very much like you, and you each know that you rolled six if and only if the other did not. Of course, still 1/2; no new relevant information here.

ETA: Say we take the limit as N goes to infinity, with everything else kept constant. We can end up with two countably infinite sets of twins, and apparently the same probabilities at each step, so we have a probability of 1/2 in the infinite case. Now, pretend that instead, we had X=N/3 (and the angel only tells us that, and nothing else). Then our probability of having a six is 1/3. As N approaches infinity, apparently our probability is still 1/3. But in this infinity land, we can still do the room pairing thing! There is a bijection between any two countably infinite sets. In fact, we could approach the infinite case with any ratio of sixes to non-sixes, as long as it's positive and less than one, and still end up with a bijection between the sixes and the non-sixes. Without a well-defined limit approaching the infinite case, we can produce any probability we want; limits need to be well-defined.

This is explained in Jaynes (2003). I don't have it with me, but if I recall it is in Chapter 15 or thereabouts on marginalization and other paradoxes.

Agreed with everything you say, but I don't think it addresses the main question. Suppose the angel does not say your (2), but the original (2): there is an actual countable infinity of copies of you. If I understand correctly, you are saying that probability theory breaks down under this information and cannot even address the question of how likely is your die to be 1/6. If this is so, isn't a serious problem for multiverse theories, as suggested in the next-to-last quoted paragraph?