Sleeping Beauty is an edge case where different reward structures are intuitively possible, and so people imagine different game payout structures behind the definition of “probability”. Once the payout structure is fixed, the confusion is gone. With a fixed payout structure&preference framework rewarding the number you output as “probability”, people don’t have a disagreement about what is the best number to output. Sleeping beauty is about definitions.)
And still, I see posts arguing that if a tree falls on a deaf Sleeping Beauty, in a forest with no one to hear it, it surely doesn’t produce a sound, because here’s how humans perceive sounds, which is the definition of a sound, and there are demonstrably no humans around the tree. (Or maybe that it surely produces the sound because here’s the physics of the sound waves, and the tree surely abides by the laws of physics, and there are demonstrably sound waves.)
This is arguing about definitions. You feel strongly that “probability” is that thing that triggers the “probability” concept neuron in your brain. If people have a different concept triggering “this is probability”, you feel like they must be wrong, because they’re pointing at something they say is a sound and you say isn’t.
Probability is something defined in math by necessity. There’s only one way to do it to not get exploited in natural betting schemes/reward structures that everyone accepts when there are no anthropics involved. But if there are multiple copies of the agent, there’s no longer a single possible betting scheme defining a single possible “probability”, and people draw the boundary/generalise differently in this situation.
You all should just call these two probabilities two different words instead of arguing which one is the correct definition for “probability”.
To be frank, it feels as if you didn’t read any of my posts on Sleeping Beauty before writing this comment. That you are simply annoyed when people arguing about substantionless semantics—and, believe me, I sympathise enourmously! - assume that I’m doing the same, based on shallow pattern matching “talks about Sleeping Beauty → semantic disagreement” and spill your annoyance at me, without validating whether your previous assumption is actually correct.
Which is a shame, because I’ve designed this whole series of posts with people like you in mind. Someone who starts from the assumption that there are two valid answers, because it was the assumption I myself used to be quite sympathetic to until I actually went forth and checked.
If it’s indeed the case, please start here and then I’d appreciate if you actually engaged with the points I made, because that post addresses the kind of criticism you are making here.
If you actually read all my Sleeping Beauty posts, saw me highlight the very specific mathematical disagreements between halfers and thirders and how utterly ungrounded the idea of using probability theory with “centred possible words” is, I don’t really understand how this kind of appealing to both sides still having a point can be a valid response.
Anyway, I’m going to address you comment step by step.
Sleeping Beauty is an edge case where different reward structures are intuitively possible
Different reward structures are possible in any probability theory problem. Make a bet on a coin toss but if the outcome is Tails—this bet is repeated three times and if it’s Heads you get punched in the face—is a completely possible reward structure for a simple coin toss problem. Is it not very intuitive? Granted, but this is besides the point. Mathematical rules are supposed to always work, even in non-intuitive cases.
Once the payout structure is fixed, the confusion is gone.
People should agree on which bets to make—this is true and this is exactly what I show in the first part of this post. But the mathematical concept of “probability” is not just about bets—which I talk about in the middle part of this post. A huge part of the confusion is still very much present. Or so it was, until I actually resolved it in the previous post.
Sleeping beauty is about definitions.
There definetely is a semantic component in the disagreement betwen halfers and thirders. But it’s the least interesting one and that’s why I’m postponing the talk about it until the next post.
The thing, you seem to be missing, is that there is also a real objective disagreement which is obfuscated by the semantic one. People noticed that halfers and thirders use different definitions and come to the conclusion that semantics is all there is and decided not to look further. But they totally should have.
My last two posts are talking about this objective matters disagreements. Is there an update on awakening or is there not? There is a disagreement about it even between thirders who, apparently agree on the definition of “probability”. Are the ways halfers and thirders define probability formally correct? It’s a strictly defined mathematical concept, mind you, not some similarity cluster category border like “sound”. Are Tails&Monday and Tails&Tuesday mutually exclusive events? You can’t just define mutual exclusivity however you like.
Probability is something defined in math by necessity.
Probability is a measure function over an event space. And if for some mathematical reasons you can’t construct an event space, your “probability” is illdefined.
You all should just call these two probabilities two different words instead of arguing which one is the correct definition for “probability”.
I’m doing both. I’ve shown that only one thing formally is probability, and in the next post I’m going to define the other thing and explore it’s properties.
I read the beginning and skimmed through the rest of the linked post. It is what I expected it to be.
We are talking about “probability”—a mathematical concept with a quite precise definition. How come we still have ambiguity about it?
Reading E.T. Jayne’s might help.
Probability is what you get as a result of some natural desiderata related to payoff structures. When anthropics are involved, there are multiple ways to extend the desiderata, that produce different numbers that you should say, depending on what you get paid for/what you care about, and accordingly different math. When there’s only a single copy of you, there’s only one kind of function, and everyone agrees on a function and then strictly defines it. When there are multiple copies of you, there are multiple possible ways you can be paid for having a number that represents something about the reality, and different generalisations of probability are possible.
Probability is what you get as a result of some natural desiderata related to payoff structures.
This is a vague gesture to a similarity cluster and not an actual definition. Remove fancy words and you end up with “Probability has something to do with betting”. Yes it does. In this post I even specify exactly what it does. You don’t need to read E.T. Jayne’s to discover this revelation. The definition of expected utility is much more helpful.
When anthropics are involved, there are multiple ways to extend the desiderata, that produce different numbers that you should say, depending on what you get paid for/what you care about, and accordingly different math.
There are always multiple ways to “extend the desiderata”. But more importantly, you don’t have to say different probability estimates depending on what you get paid for/what you care about. This is the exact kind of nonsense that I’m calling out in this post. Probabilities are about what evidence you have. Utilities are about what you care about. You don’t need to use thirder probabilities for per awakening betting. Do you disagree with me here?
When there’s only a single copy of you, there’s only one kind of function, and everyone agrees on a function and then strictly defines it. When there are multiple copies of you, there are multiple possible ways you can be paid for having a number that represents something about the reality, and different generalisations of probability are possible.
How is it different from talking about probability of a specific person to observe an event and probability of any person from a group to observe an event? The fact that people from the group are exact copies doesn’t suddenly makes anthropics a separate magisteria.
Moreover, there are no independent copies in Sleeping Beauty. On Tails, there are two sequential time states. The fact that people are trying to make a sample space out of them directly contradicts its definition.
When we are talking just about betting, one can always come up with its own functions, it’s own way to separate expected utility of an event into “utility” and “probability”. But then their “utilities” will be constantly shifting due to receiving new evidence and “probabilities” will occasionally ignore new evidence, and shift for other reasons. And pointing at this kind of weird behavior is a completely reasonable reaction. Can a person still use such definitions consistently? Sure. But this is not a way to carve reality by its joints. And I’m not just talking about betting. I specifically wrote a whole post about fundamental mathematical reasons, before starting talking about it.
Sleeping Beauty is an edge case where different reward structures are intuitively possible, and so people imagine different game payout structures behind the definition of “probability”. Once the payout structure is fixed, the confusion is gone. With a fixed payout structure&preference framework rewarding the number you output as “probability”, people don’t have a disagreement about what is the best number to output. Sleeping beauty is about definitions.)
And still, I see posts arguing that if a tree falls on a deaf Sleeping Beauty, in a forest with no one to hear it, it surely doesn’t produce a sound, because here’s how humans perceive sounds, which is the definition of a sound, and there are demonstrably no humans around the tree. (Or maybe that it surely produces the sound because here’s the physics of the sound waves, and the tree surely abides by the laws of physics, and there are demonstrably sound waves.)
This is arguing about definitions. You feel strongly that “probability” is that thing that triggers the “probability” concept neuron in your brain. If people have a different concept triggering “this is probability”, you feel like they must be wrong, because they’re pointing at something they say is a sound and you say isn’t.
Probability is something defined in math by necessity. There’s only one way to do it to not get exploited in natural betting schemes/reward structures that everyone accepts when there are no anthropics involved. But if there are multiple copies of the agent, there’s no longer a single possible betting scheme defining a single possible “probability”, and people draw the boundary/generalise differently in this situation.
You all should just call these two probabilities two different words instead of arguing which one is the correct definition for “probability”.
To be frank, it feels as if you didn’t read any of my posts on Sleeping Beauty before writing this comment. That you are simply annoyed when people arguing about substantionless semantics—and, believe me, I sympathise enourmously! - assume that I’m doing the same, based on shallow pattern matching “talks about Sleeping Beauty → semantic disagreement” and spill your annoyance at me, without validating whether your previous assumption is actually correct.
Which is a shame, because I’ve designed this whole series of posts with people like you in mind. Someone who starts from the assumption that there are two valid answers, because it was the assumption I myself used to be quite sympathetic to until I actually went forth and checked.
If it’s indeed the case, please start here and then I’d appreciate if you actually engaged with the points I made, because that post addresses the kind of criticism you are making here.
If you actually read all my Sleeping Beauty posts, saw me highlight the very specific mathematical disagreements between halfers and thirders and how utterly ungrounded the idea of using probability theory with “centred possible words” is, I don’t really understand how this kind of appealing to both sides still having a point can be a valid response.
Anyway, I’m going to address you comment step by step.
Different reward structures are possible in any probability theory problem. Make a bet on a coin toss but if the outcome is Tails—this bet is repeated three times and if it’s Heads you get punched in the face—is a completely possible reward structure for a simple coin toss problem. Is it not very intuitive? Granted, but this is besides the point. Mathematical rules are supposed to always work, even in non-intuitive cases.
People should agree on which bets to make—this is true and this is exactly what I show in the first part of this post. But the mathematical concept of “probability” is not just about bets—which I talk about in the middle part of this post. A huge part of the confusion is still very much present. Or so it was, until I actually resolved it in the previous post.
There definetely is a semantic component in the disagreement betwen halfers and thirders. But it’s the least interesting one and that’s why I’m postponing the talk about it until the next post.
The thing, you seem to be missing, is that there is also a real objective disagreement which is obfuscated by the semantic one. People noticed that halfers and thirders use different definitions and come to the conclusion that semantics is all there is and decided not to look further. But they totally should have.
My last two posts are talking about this objective matters disagreements. Is there an update on awakening or is there not? There is a disagreement about it even between thirders who, apparently agree on the definition of “probability”. Are the ways halfers and thirders define probability formally correct? It’s a strictly defined mathematical concept, mind you, not some similarity cluster category border like “sound”. Are Tails&Monday and Tails&Tuesday mutually exclusive events? You can’t just define mutual exclusivity however you like.
Probability is a measure function over an event space. And if for some mathematical reasons you can’t construct an event space, your “probability” is illdefined.
I’m doing both. I’ve shown that only one thing formally is probability, and in the next post I’m going to define the other thing and explore it’s properties.
I read the beginning and skimmed through the rest of the linked post. It is what I expected it to be.
Reading E.T. Jayne’s might help.
Probability is what you get as a result of some natural desiderata related to payoff structures. When anthropics are involved, there are multiple ways to extend the desiderata, that produce different numbers that you should say, depending on what you get paid for/what you care about, and accordingly different math. When there’s only a single copy of you, there’s only one kind of function, and everyone agrees on a function and then strictly defines it. When there are multiple copies of you, there are multiple possible ways you can be paid for having a number that represents something about the reality, and different generalisations of probability are possible.
This is surprising to me. Are you up to a a more detailed discussion? What do you think about the statistical analysis and the debunk of centred possible worlds? I haven’t seen these points being raised or addressed before and they are definitely not about semantics. The fact that sequential events are not mutually exclusive can be formally proven. It’s not a matter of perspective at all! We could use the dialogues feature, if you’d like.
This is a vague gesture to a similarity cluster and not an actual definition. Remove fancy words and you end up with “Probability has something to do with betting”. Yes it does. In this post I even specify exactly what it does. You don’t need to read E.T. Jayne’s to discover this revelation. The definition of expected utility is much more helpful.
There are always multiple ways to “extend the desiderata”. But more importantly, you don’t have to say different probability estimates depending on what you get paid for/what you care about. This is the exact kind of nonsense that I’m calling out in this post. Probabilities are about what evidence you have. Utilities are about what you care about. You don’t need to use thirder probabilities for per awakening betting. Do you disagree with me here?
How is it different from talking about probability of a specific person to observe an event and probability of any person from a group to observe an event? The fact that people from the group are exact copies doesn’t suddenly makes anthropics a separate magisteria.
Moreover, there are no independent copies in Sleeping Beauty. On Tails, there are two sequential time states. The fact that people are trying to make a sample space out of them directly contradicts its definition.
When we are talking just about betting, one can always come up with its own functions, it’s own way to separate expected utility of an event into “utility” and “probability”. But then their “utilities” will be constantly shifting due to receiving new evidence and “probabilities” will occasionally ignore new evidence, and shift for other reasons. And pointing at this kind of weird behavior is a completely reasonable reaction. Can a person still use such definitions consistently? Sure. But this is not a way to carve reality by its joints. And I’m not just talking about betting. I specifically wrote a whole post about fundamental mathematical reasons, before starting talking about it.