For the outside view: Imagine that an outside observer uses a fair coin to observe one of two rooms (assuming merging in the red room has happened). They will observe either a red room or a green room, with a copy in each. However, the observer who was copied has different chances of observing the green and red rooms. Even if the outside observer has access to the entire current state of the world (but not the character of mixing of the paths in the past), they can’t determine the copied observer’s subjective chances. This implies that subjective unmeasurable probabilities are real.
Even without merging, an outside observer will observe three rooms with equal 1⁄3 probability for each, while an insider will observe room 1 with 1⁄2 probability. In cases of multiple sequential copying events, the subjective probability for the last copy becomes extremely small, making the difference between outside and inside perspectives significant.
When I spoke about the similarity with the Sleeping Beauty problem, I meant its typical interpretation. It’s an important contribution to recognize that Monday-tails and Tuesday-tails are not independent events.
However, I have an impression that this may result in a paradoxical two-thirder solution: In it, Sleeping Beauty updates only once – recognizing that there are two more chances to be in tails. But she doesn’t update again upon knowing it’s Monday, as Monday-tails and Tuesday-tails are the same event. In that case, despite knowing it’s Monday, she maintains a 2⁄3 credence that she’s in the tails world. This is technically equivalent to the ‘future anthropic shadow’ or anti-doomsday argument – the belief that one is now in the world with the longest possible survival.
Imagine that an outside observer uses a fair coin to observe one of two rooms (assuming merging in the red room has happened). They will observe either a red room or a green room, with a copy in each. However, the observer who was copied has different chances of observing the green and red rooms.
Well obviously. The observer and the person being copied participate in non-isomorphic experiments with different sampling. There is nothing surprising about it. On the other hand, if we make the experiments isomorphic:
Two coins are tossed and the observer is brought into the green room if both are Heads, and is brought to the red room, otherwise
Then both the observer and the person being copied will have the same probabilities.
Even without merging, an outside observer will observe three rooms with equal 1⁄3 probability for each, while an insider will observe room 1 with 1⁄2 probability.
Likewise, nothing is preventing you from designing an experimental setting where an observer have 1⁄2 probability for room 1 just as the person who is being copied.
When I spoke about the similarity with the Sleeping Beauty problem, I meant its typical interpretation.
I’m not sure what use is investigating a wrong interpretation. It’s a common confusion that one has to reason about problems involving amnesia the same way as about problems involving copying. Everyone just seem to assume it for no particular reason and therefore got stuck.
However, I have an impression that this may result in a paradoxical two-thirder solution: In it, Sleeping Beauty updates only once – recognizing that there are two more chances to be in tails. But she doesn’t update again upon knowing it’s Monday, as Monday-tails and Tuesday-tails are the same event. In that case, despite knowing it’s Monday, she maintains a 2⁄3 credence that she’s in the tails world.
This seems to be the worst of both worlds. Not only you update on a completely expected event, you then keep this estimate, expecting to be able to guess a future coin toss better than chance. An obvious way to lose all your money via betting.
For the outside view: Imagine that an outside observer uses a fair coin to observe one of two rooms (assuming merging in the red room has happened). They will observe either a red room or a green room, with a copy in each. However, the observer who was copied has different chances of observing the green and red rooms. Even if the outside observer has access to the entire current state of the world (but not the character of mixing of the paths in the past), they can’t determine the copied observer’s subjective chances. This implies that subjective unmeasurable probabilities are real.
Even without merging, an outside observer will observe three rooms with equal 1⁄3 probability for each, while an insider will observe room 1 with 1⁄2 probability. In cases of multiple sequential copying events, the subjective probability for the last copy becomes extremely small, making the difference between outside and inside perspectives significant.
When I spoke about the similarity with the Sleeping Beauty problem, I meant its typical interpretation. It’s an important contribution to recognize that Monday-tails and Tuesday-tails are not independent events.
However, I have an impression that this may result in a paradoxical two-thirder solution: In it, Sleeping Beauty updates only once – recognizing that there are two more chances to be in tails. But she doesn’t update again upon knowing it’s Monday, as Monday-tails and Tuesday-tails are the same event. In that case, despite knowing it’s Monday, she maintains a 2⁄3 credence that she’s in the tails world. This is technically equivalent to the ‘future anthropic shadow’ or anti-doomsday argument – the belief that one is now in the world with the longest possible survival.
Well obviously. The observer and the person being copied participate in non-isomorphic experiments with different sampling. There is nothing surprising about it. On the other hand, if we make the experiments isomorphic:
Two coins are tossed and the observer is brought into the green room if both are Heads, and is brought to the red room, otherwise
Then both the observer and the person being copied will have the same probabilities.
Likewise, nothing is preventing you from designing an experimental setting where an observer have 1⁄2 probability for room 1 just as the person who is being copied.
I’m not sure what use is investigating a wrong interpretation. It’s a common confusion that one has to reason about problems involving amnesia the same way as about problems involving copying. Everyone just seem to assume it for no particular reason and therefore got stuck.
This seems to be the worst of both worlds. Not only you update on a completely expected event, you then keep this estimate, expecting to be able to guess a future coin toss better than chance. An obvious way to lose all your money via betting.