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.

And the same holds for the other guy?

Yes, it does.

It is important to remember that first we rolled dice, and only later the angel decided that we two will be a pair. If both of us would not roll 6, the angel would simply decide to not bring us together. -- The paradox of infinity is that the angel will always have enough other people to bring, if we both happen to not roll 6.

Let's make it simple by removing the infinities.

There are exactly one million people, each of them rolls a die. An angel blindfolds them, and sorts them into two large groups: those who rolled a six, and those who didn't. The blindfolded people cannot compare the sizes of their groups. Then the angel asks you: "What is the probability of you having rolled six?" (You can assume that the angel speaks simultaneously with everyone, so there is nothing special about asking you this question, instead of asking it someone else.)

A logical answer would be: 1/6.

Then the angel says: "Now I choose a random person from the other group; let's call him Joe. What is the probability that Joe has rolled a six?"

A logical answer: 5/6.

The angel says: "However, Joe believes his chance of having rolled a six is 1/6, and he uses the same reasoning that you do. Is he wrong or what? Remember that I am having this dialogue simultaneously with each participant of the experiment."

A logical answer: Probability is in the mind. In my mind, Joe has a probability 5/6 of having rolled a six. In Joe's mind, he has a probability 1/6 of having rolled a six. These are two different questions.

Imagine that there are six people: me, Joe, Adam, Betty, Cecil, Daniel; and exactly one of us has rolled a 6. If it was me who rolled the six, you could have chosen any of {Joe, Adam, Betty, Cecil, Daniel} to be my twin. But if it was Joe who rolled the six, you didn't have another choice for my twin, only Joe. So the fact that Joe is my twin, is for me an evidence that Joe has rolled the six. And symetrically, if you chose me as a twin for Joe, that is for him an evidence that I have rolled the six.

Because we have this finite scenario, one person was chosen five times as a "twin", one person was chosen once, and four people we not chosen as "twins" at all. So we have only one of six people rolling 6, but we have five of six "twins" rolling 6. Therefore my probability of rolling 6 is 1/6 and my "twin"'s probability is 5/6.

...however in the infinite version, even if one subset is 5 times smaller than the other, the angel can choose a twin for everyone. But in some sense it means that if you rolled six, you are five times more likely to be selected as someone's pair, as you would be if the angel just ignored the numbers on dice and paired people randomly.

In my mind, Joe has a probability 5/6 of having rolled a six. In Joe's mind, he has a probability 1/6 of having rolled a six. These are two different questions.

While they are two different questions, you both have the same evidence and the same priors. They are symmetric questions, and should have identical answers.

0Thomas13y

So, I have 1/6 probability for the 6, he has 1/6 probability for the 6 and one of us has the 6 for sure?