I think that’s a good summary of where our disagreement lies. I think that your “sample worlds until the sky turns out blue” methodology for generating a sample is very different to the existence/non-existence case, especially if there is actually only one world! If there are many worlds, it’s more similar, and this is why I think anthropic shadow has more of a chance of working in that case (that was my ‘Possible Solution #2’).
I find it very interesting that your intuition on the Russian roulette is the other way round to mine. So if there are two guns, one with 1/1000 probability of firing, and one with 999/1000 probability of firing, and you pick one at random and it doesn’t fire, you think that you have no information about which gun you picked? Because you’d be dead otherwise?
I agree that we don’t get very far by just stating our different intuitions, so let me try to convince you of my point of view a different way:
Suppose that you really do have no information after firing a gun once and surviving. Then, if told to play the game again, you should be indifferent between sticking with the same gun, or switching to the different gun. Lets say you settle on the switching strategy (maybe I offer you some trivial incentive to do so). I, on the other hand, would strongly favour sticking with the same gun. This is because I think I have extremely strong evidence that the gun I picked is the less risky one, if I have survived once.
Now lets take a birds-eye view, and imagine an outside observer watching the game, betting on which one of us is more likely to survive through two rounds. Obviously they would favour me over you. My odds of survival are approximately 50% (it more or less just depends on whether I pick the safe gun first or not). Your odds of survival are approximately 1 in 1000 (you are guaranteed to have one shot with the dangerous gun).
This doesn’t prove that your approach to formulating probabilities is wrong, but if ultimately we are interested in using probabilities to inform our decisions, I think this suggests that my approach is better.
On the fine tuning, if it is different, I would like to understand why. I’d love to know what the general procedure we’re supposed to use is to analyse anthropic problems. At the moment I struggle to see how it could both include the anthropic shadow effect, and also have the fine tuning of cosmological constants be taken as evidence for a multiverse.
Here’s a rough sketch of how we could, potentially, think about anthropic problems. Let Pt be a sequence of true, bird’s-eye view probability measures and Qt your own measures, trying to mimic Pt as closely as possible. These measures aren’t defined on the same sigma-algebra. The sequence of true measures is defined on some original sigma-algebra Σ, but your measure is defined only on the sigma-algebra {A∩{ω where the sky is blue at time t}}.
Now, the best-known probability measure defined on this set is the conditional probability
Qt(A)=Pt(A∣ where the sky is blue at time t).
This is, in a sense, the probability measure that most closely mimics Pt . On the other hand, the measure that mimics Pt most closely is Qt(A)=Pt(A), hands down. This measure has a problem though, namely that maxQt(A)<1, hence it isn’t a probability measure anymore.
I think the main reason why I intuitively want to condition on the color of the sky is that I want to work with proper probability measures, not just measures bounded by 0 and 1. (That’s why I’m talking about, e.g., being “uncomfortable pretending we could have observed non-existence”.) But your end goal is to have the best measure on the data you can actually observe, taking into account possibilities you can’t observe. This naturally leads us to Qt(A)=Pt(A) instead of Qt(A)=Qt(A)=Pt(A∣ where the sky is blue at time t).
There is another independent aspect to anthropic reasoning too, which is how you assign probabilities to ‘indexical’ facts. This is the part of anthropic reasoning I always thought was more contentious. For example, if two people are created, one with red hair and one with blue hair, and you are one of these people, what is the probability that you have red hair (before you look in the mirror)? We are supposed to use the ‘Self-Sampling Assumption’ here, and say the answer is 1⁄2, but if you just naively apply that rule too widely then you can end up with conclusions like the Doomsday Argument, or Adam+Eve paradox.
I think that a complete account of anthropic reasoning would need to cover this as well, but I think what you’ve outlined is a good summary of how we should treat cases where we are only able to observe certain outcomes because we do not exist in others.
The roulette example might get to the heart of the problem with the worm’s-eye view! From the worm’s-eye view, the sky will always be blue, so P(skycolor=green)=0, making it impossible to deal with problems where the sky might turn green in the future.
In the roulette example, we’re effectively dealing with an expected utility problem where we condition on existence when learning about the probability, but not when we act. That looks incoherent to me; we can’t condition and uncondition on an event willy-nilly: Either we will live in a world where an event must be true, or we don’t. So yeah, it seems like you’re right, and we’re effectively treating existence as a certainty when looking at the problem from the worm’s-eye view.
As I see it, this strongly suggests we should take the bird’s-eye view, as you proposed, and not the worm’s-eye view. Or something else entirely; I’m still uncomfortable pretending we could have observed non-existence.
I think that’s a good summary of where our disagreement lies. I think that your “sample worlds until the sky turns out blue” methodology for generating a sample is very different to the existence/non-existence case, especially if there is actually only one world! If there are many worlds, it’s more similar, and this is why I think anthropic shadow has more of a chance of working in that case (that was my ‘Possible Solution #2’).
I find it very interesting that your intuition on the Russian roulette is the other way round to mine. So if there are two guns, one with 1/1000 probability of firing, and one with 999/1000 probability of firing, and you pick one at random and it doesn’t fire, you think that you have no information about which gun you picked? Because you’d be dead otherwise?
I agree that we don’t get very far by just stating our different intuitions, so let me try to convince you of my point of view a different way:
Suppose that you really do have no information after firing a gun once and surviving. Then, if told to play the game again, you should be indifferent between sticking with the same gun, or switching to the different gun. Lets say you settle on the switching strategy (maybe I offer you some trivial incentive to do so). I, on the other hand, would strongly favour sticking with the same gun. This is because I think I have extremely strong evidence that the gun I picked is the less risky one, if I have survived once.
Now lets take a birds-eye view, and imagine an outside observer watching the game, betting on which one of us is more likely to survive through two rounds. Obviously they would favour me over you. My odds of survival are approximately 50% (it more or less just depends on whether I pick the safe gun first or not). Your odds of survival are approximately 1 in 1000 (you are guaranteed to have one shot with the dangerous gun).
This doesn’t prove that your approach to formulating probabilities is wrong, but if ultimately we are interested in using probabilities to inform our decisions, I think this suggests that my approach is better.
On the fine tuning, if it is different, I would like to understand why. I’d love to know what the general procedure we’re supposed to use is to analyse anthropic problems. At the moment I struggle to see how it could both include the anthropic shadow effect, and also have the fine tuning of cosmological constants be taken as evidence for a multiverse.
Here’s a rough sketch of how we could, potentially, think about anthropic problems. Let Pt be a sequence of true, bird’s-eye view probability measures and Qt your own measures, trying to mimic Pt as closely as possible. These measures aren’t defined on the same sigma-algebra. The sequence of true measures is defined on some original sigma-algebra Σ, but your measure is defined only on the sigma-algebra {A∩{ω where the sky is blue at time t}}.
Qt(A)=Pt(A∣ where the sky is blue at time t).Now, the best-known probability measure defined on this set is the conditional probability
This is, in a sense, the probability measure that most closely mimics Pt . On the other hand, the measure that mimics Pt most closely is Qt(A)=Pt(A), hands down. This measure has a problem though, namely that maxQt(A)<1, hence it isn’t a probability measure anymore.
I think the main reason why I intuitively want to condition on the color of the sky is that I want to work with proper probability measures, not just measures bounded by 0 and 1. (That’s why I’m talking about, e.g., being “uncomfortable pretending we could have observed non-existence”.) But your end goal is to have the best measure on the data you can actually observe, taking into account possibilities you can’t observe. This naturally leads us to Qt(A)=Pt(A) instead of Qt(A)=Qt(A)=Pt(A∣ where the sky is blue at time t).
I think that makes sense!
There is another independent aspect to anthropic reasoning too, which is how you assign probabilities to ‘indexical’ facts. This is the part of anthropic reasoning I always thought was more contentious. For example, if two people are created, one with red hair and one with blue hair, and you are one of these people, what is the probability that you have red hair (before you look in the mirror)? We are supposed to use the ‘Self-Sampling Assumption’ here, and say the answer is 1⁄2, but if you just naively apply that rule too widely then you can end up with conclusions like the Doomsday Argument, or Adam+Eve paradox.
I think that a complete account of anthropic reasoning would need to cover this as well, but I think what you’ve outlined is a good summary of how we should treat cases where we are only able to observe certain outcomes because we do not exist in others.
The roulette example might get to the heart of the problem with the worm’s-eye view! From the worm’s-eye view, the sky will always be blue, so P(skycolor=green)=0, making it impossible to deal with problems where the sky might turn green in the future.
In the roulette example, we’re effectively dealing with an expected utility problem where we condition on existence when learning about the probability, but not when we act. That looks incoherent to me; we can’t condition and uncondition on an event willy-nilly: Either we will live in a world where an event must be true, or we don’t. So yeah, it seems like you’re right, and we’re effectively treating existence as a certainty when looking at the problem from the worm’s-eye view.
As I see it, this strongly suggests we should take the bird’s-eye view, as you proposed, and not the worm’s-eye view. Or something else entirely; I’m still uncomfortable pretending we could have observed non-existence.