I personally think that the only “good” definition (I’ll specify this more at the end) is that a probability of 14 should occur one in four times in the relevant reference class. I’ve previously called this view “generalized frequentism”, where we use the idea of repeated experiments to define probabilities, but generalizes the notion of “experiment” to subsume all instances of an agent with incomplete information acting in the real world (hence subsuming the definition as subjective confidence).
Why do you suddenly substitute the notion of “probability experiment” with the notion of “reference class”? What do you achieve by this?
From my perspective, this is where the source of confusion lingers. Probability experiment can be precisely specified: the description of any probability theory problem is supposed to be that. But “reference class” is misleading and up for the interpretation.
There are difficulties here with defining the reference class, but I think they can be adequately addressed, and anyway, those don’t matter for the sleeping beauty experiment because there, the reference classes is actually really straight-forward. Among the times that you as an agent are participating in the experiment and are woken up and interviewed (and are called Sleeping Beauty, if you want to include this in the reference class), one third will have the coin heads, so the probability is 13.
And indeed, because of this “reference class” business you suddenly started treating individual awakening of Sleeping Beauty as mutually exclusive outcome, even though it’s absolutely not the case in the experiment as stated. I don’t see how you would make such mistake if you kept using the term “probability experiment” without switching to speculate about “reference classes”.
Among the iterations of Sleeping Beauty probability experiment that a participant awakes, half the time the coin is Heads so the probability is 1⁄2.
EDIT: @ProgramCrafter the description of the experiment clearly states that that when the coin is Tails the Beauty is to be awaken twice in the same iteration of the experiment. Therefore, individual awakennings are not mutually exclisive with each other: more than one can happen in the same iteration of the experiment.
Why do you suddenly substitute the notion of “probability experiment” with the notion of “reference class”? What do you achieve by this?
Just to be clear, the reference class here is the set of all instances across all of space and time where an agent is in the same “situation” as you (where the thing you can argue about is how precisely one has to specify the situation). So in the case of the coinflip, it’s all instances across space and time where you flip a physical coin (plus, if you want to specify further, any number of other details about the current situation).
So with that said, to answer your question: why define probabilities in terms of this concept? Because I don’t think I want a definition of probability that doesn’t align with this view, when it’s applicable. If we can discretely count the number of instances across the history of the universe that fit the current situation , and we know some event happens in one third of those instances, then I think the probability has to be one third. This seems very self-evident to me; it seems exactly what the concept of probability is supposed to do.
I guess one analogy—suppose one third of all houses is painted blue from the outside and one third red, and you’re in one house but have no idea which one. What’s the probability that it’s blue? I think it’s 2⁄3, and I think this situation is precisely analogous to the reference class construction. Like I actually think there is no relevant difference; you’re in one of the situations that fit the current situation (trivially so), and you can’t tell which one (by construction; if you could, that would be included in the definition of the reference class, which would make it different from the others). Again, this just seems to get at precisely the core of what a probability should do.
So I think that answers it? Like I said, I think you can define “probability” differently, but if the probability doesn’t align with reference class counting, then it seems to me that the point of the concept has been lost. (And if you do agree with that, the question is just whether or not reference class counting is applicable, which I haven’t really justified in my reply, but for Sleeping Beauty it seems straight-forward.)
So with that said, to answer your question: why define probabilities in terms of this concept? Because I don’t think I want a definition of probability that doesn’t align with this view, when it’s applicable.
Suppose I want matrix multiplication to be commutative. Surely it would be so convinient if it was! I can define some operator * over matrixes so that A*B = B*A. I can even call this operator “matrix multiplication”.
But did I just make matrix multiplication, as it’s conventionally defined, commutative? Of course not. I logically pinpointed a new function and called it the same way as the previous function is being called, but it didn’t change anything about how the previous function is logically pinpointed.
My new function may have some interesting applications and therefore deserve to be talked about in its own right. But calling it’s “matrix multiplication” is very misleading. And if I were to participate in conversation about matrix multiplication while talking about my function I’d be confusing everyone.
This is basically the situation that we have here.
Initially probability function is defined over iterations of probability experiment. You define a different function over all space and time, which you still call “probability”. It surely has properties that you like, but it’s a different function! Please use another name, this is already taken. Or add a disclaimer. Preferably do both. You know how easy it is to confuse people with such things! Definetely, do not start participating in the conversations about probability while talking about your function.
If we can discretely count the number of instances across the history of the universe that fit the current situation , and we know some event happens in one third of those instances, then I think the probability has to be one third. This seems very self-evident to me; it seems exactly what the concept of probability is supposed to do.
I guess one analogy—suppose one third of all houses is painted blue from the outside and one third red, and you’re in one house but have no idea which one. What’s the probability that it’s blue?
As long as these instances are independent of each other—sure. Like with your houses analogy. When we are dealing with simple, central cases there is no diasagreement between probability and weighted probability and so nothing to argue about.
But as soon as we are dealing with more complicated scenario where there is no independence and it’s possible to be inside multiple houses in the same instance… Surely, you see how demanding to have coherent P(Red xor Blue) becomes unfeasible?
The problem is, our intuitions are too eager to assume that everything as independent. We are used to think in terms of physical time, using our memory as something that allows us to orient in it. This is why amnesia scenarios are so mindboggling to us!
And that’s why the notion of probability experiment where every single trial is independent and the outcomes in any single trial are mutually exclusive is so important. We strictly define what the “situation” means and therefore do not allow ourselves to be tricked. We can clearly see that individual awakenings can’t be treated as outcomes of the Sleeping Beauty experiment.
But when you are thinking in terms of “reference classes” your definition of “situation” is too vague. And so you allow yourself to count the same house multiple times. Treat yourself not as a person participating in the experiment but as an “awakening state of the person”, even though one awakening state necessary follows the other.
if the probability doesn’t align with reference class counting, then it seems to me that the point of the concept has been lost.
The “point of probability” is lost when it doesn’t allign with reasoning about instances of probability experiments. Namely, we are starting to talk about something else, instead of what was logically pinpointed as probability in the first place. Most of the time reasoning about reference classes does allign with it, so you do not notice the difference. But once in a while it doesn’t and so you end up having “probability” that contradicts conservation of expected evidence and “utility” shifting back and forth.
So what’s the point of these reference classes? What’s so valuable in them? As far as I can see they do not bring anything to the table except extra confusion.
Upon rereading your posts, I retract disagreement on “mutually exclusive outcomes”. Instead...
Initially probability function is defined over iterations of probability experiment. You define a different function over all space and time, which you still call “probability”. It surely has properties that you like, but it’s a different function! Please use another name, this is already taken. Or add a disclaimer. Preferably do both. You know how easy it is to confuse people with such things! Definetely, do not start participating in the conversations about probability while talking about your function.
An obvious way to do so is put a hazard sign on “probability” and just not use it, not putting resources into figuring out what “probability” SB should name, isn’t it? For instance, suppose Sleeping Beauty claims “my credence for Tails is 1π”; any specific objection would be based on what you expected to hear.
(And now I realize a possible point why you’re arguing to keep “probability” term for such scenarios well-defined; so that people in ~anthropic settings can tell you their probability estimates and you, being observer, could update on that information.)
As for why I believe probability theory to be useful in life despite the fact that sometimes different tools need to be used: I believe disappearing as a Boltzmann brain or simulated person is balanced out by appearing the same way, dissolving into different quantum branches is balanced out by branches reassembling, and likewise for most processes.
An obvious way to do so is put a hazard sign on “probability” and just not use it, not putting resources into figuring out what “probability” SB should name, isn’t it?
It’s an obvious thing to do when dealing with simularity clusters poorly defined in natural language. Not so much, when we are talking about a logically pinpointed mathematical concept which we know are crucial for epistemology.
(And now I realize a possible point why you’re arguing to keep “probability” term for such scenarios well-defined; so that people in ~anthropic settings can tell you their probability estimates and you, being observer, could update on that information.)
It’s not just about anthropic scenarios and not just about me being able to understand other people. It’s about general truth preserving mechanism of logical and mathematical reasoning. When people just use different definitions—this is annoying but fine. But when they use different definitions without realizing that these definitions are different and, moreover insist that it’s you who is making a mistake—then we have an actual disagreement about math which will provide more confusion along the way. Anthropic scenarious are just the ones where this confusion is noticeable.
As for why I believe probability theory to be useful in life despite the fact that sometimes different tools need to be used
What exactly do you mean by “different tools need to be used”? Can you give me an example?
What exactly do you mean by “different tools need to be used”? Can you give me an example?
I mean that Beauty should maintain full model of experiment, and use decision theory as well as probability theory (if latter is even useful, which it admittedly seems to be). If she didn’t keep track of full setup but only “a fair coin was flipped, so the odds are 1:1”, she would predictably lose when betting on the coin outcome.
Also, I’ve minted another “paradox” version. I can predict you’ll take issue with one of formulations in it, but what do you think about it?
A fair coin is flipped, hidden from you.
On Heads, you’re waken up on Monday, asked “what credence do you have that coin landed Heads?”; on Tuesday, you’re let go.
If coin landed Tails, you’re waken up on Monday and still asked “what credence do you have that coin landed Heads?”; then, with no memory erasure, you’re waken up on Tuesday, and experimenter says to you: “Name the credence that coin landed Heads, but you must name the exact same number as yesterday”. Afterwards, you’re let go.
If you don’t follow experiment protocol, you lose/lose out on some reward.
I suppose the participant is just supposed to lie about their credence here in order to “win”.
On Tuesday your credence in Heads supposed to be 0, but saying the true value would go against the experimental protocol unless you also said that your credence is 0 on Monday, which would also be a lie.
She certainly gets a reward for following experimental protocol, but beyond that… I concur there’s the problem, and I have the same issue with standard formulation asking for probability.
In particular, pushing problem out to morality “what should Sleeping Beauty answer so that she doesn’t feel as if she’s lying” doesn’t solve anything either; rather, it feels like asking question “is continuum hypothesis true?” providing only options ‘true’ and ‘false’, while it’s actually independent of ZFC axioms (claims of it or of its negation produce different models, neither proven to self-contradict).
P.S. One more analogue: there’s a field, and some people (experimenters) are asking whether it rained recently with clear intent to walk through if it didn’t; you know it didn’t rain but there are mines all over the field. I argue you should mention the mines first (“that probability—which by the way will be 1⁄2 - can be found out, conforms to epistemology, but isn’t directly usable anywhere”) before saying if there was rain.
As long as these instances are independent of each other—sure. Like with your houses analogy. When we are dealing with simple, central cases there is no diasagreement between probability and weighted probability and so nothing to argue about.
But as soon as we are dealing with more complicated scenario where there is no independence and it’s possible to be inside multiple houses in the same instance
If you can demonstrate how, in the reference class setting, there is a relevant criterion by which several instances should be grouped together, then I think you could have an argument.
If you look at space-time from above, there’s two blue houses for every red house. Sorry I meant there’s two SB(=Sleeping Beauty)-tails instances for every SB-heads instance. The two instances you want to group together (tails-Monday & tails-Tuesday) aren’t actually at the same time (not that I think it matters). If the universe is very large of Many Worlds is true, then there are in fact many instances of Monday-heads, Monday-tails, and Tuesday tails occurring at the same time, and I don’t think you want to group those together.
In any case, from the PoV of SB, all instances look identical to you. So by what criterion should we group some of them together? That’s the thing I think your position requires (just because you accept reference classes are a priori valid and then become invalid in some cases), and I don’t see the criterion.
Why do you suddenly substitute the notion of “probability experiment” with the notion of “reference class”? What do you achieve by this?
From my perspective, this is where the source of confusion lingers. Probability experiment can be precisely specified: the description of any probability theory problem is supposed to be that. But “reference class” is misleading and up for the interpretation.
And indeed, because of this “reference class” business you suddenly started treating individual awakening of Sleeping Beauty as mutually exclusive outcome, even though it’s absolutely not the case in the experiment as stated. I don’t see how you would make such mistake if you kept using the term “probability experiment” without switching to speculate about “reference classes”.
Among the iterations of Sleeping Beauty probability experiment that a participant awakes, half the time the coin is Heads so the probability is 1⁄2.
Here there are no difficulties to address—everything is crystal clear. You just need to calm the instinctive urge to weight the probability by the number of awakenings, which would be talking about a different mathematical concept.
EDIT: @ProgramCrafter the description of the experiment clearly states that that when the coin is Tails the Beauty is to be awaken twice in the same iteration of the experiment. Therefore, individual awakennings are not mutually exclisive with each other: more than one can happen in the same iteration of the experiment.
Just to be clear, the reference class here is the set of all instances across all of space and time where an agent is in the same “situation” as you (where the thing you can argue about is how precisely one has to specify the situation). So in the case of the coinflip, it’s all instances across space and time where you flip a physical coin (plus, if you want to specify further, any number of other details about the current situation).
So with that said, to answer your question: why define probabilities in terms of this concept? Because I don’t think I want a definition of probability that doesn’t align with this view, when it’s applicable. If we can discretely count the number of instances across the history of the universe that fit the current situation , and we know some event happens in one third of those instances, then I think the probability has to be one third. This seems very self-evident to me; it seems exactly what the concept of probability is supposed to do.
I guess one analogy—suppose one third of all houses is painted blue from the outside and one third red, and you’re in one house but have no idea which one. What’s the probability that it’s blue? I think it’s 2⁄3, and I think this situation is precisely analogous to the reference class construction. Like I actually think there is no relevant difference; you’re in one of the situations that fit the current situation (trivially so), and you can’t tell which one (by construction; if you could, that would be included in the definition of the reference class, which would make it different from the others). Again, this just seems to get at precisely the core of what a probability should do.
So I think that answers it? Like I said, I think you can define “probability” differently, but if the probability doesn’t align with reference class counting, then it seems to me that the point of the concept has been lost. (And if you do agree with that, the question is just whether or not reference class counting is applicable, which I haven’t really justified in my reply, but for Sleeping Beauty it seems straight-forward.)
Suppose I want matrix multiplication to be commutative. Surely it would be so convinient if it was! I can define some operator * over matrixes so that A*B = B*A. I can even call this operator “matrix multiplication”.
But did I just make matrix multiplication, as it’s conventionally defined, commutative? Of course not. I logically pinpointed a new function and called it the same way as the previous function is being called, but it didn’t change anything about how the previous function is logically pinpointed.
My new function may have some interesting applications and therefore deserve to be talked about in its own right. But calling it’s “matrix multiplication” is very misleading. And if I were to participate in conversation about matrix multiplication while talking about my function I’d be confusing everyone.
This is basically the situation that we have here.
Initially probability function is defined over iterations of probability experiment. You define a different function over all space and time, which you still call “probability”. It surely has properties that you like, but it’s a different function! Please use another name, this is already taken. Or add a disclaimer. Preferably do both. You know how easy it is to confuse people with such things! Definetely, do not start participating in the conversations about probability while talking about your function.
As long as these instances are independent of each other—sure. Like with your houses analogy. When we are dealing with simple, central cases there is no diasagreement between probability and weighted probability and so nothing to argue about.
But as soon as we are dealing with more complicated scenario where there is no independence and it’s possible to be inside multiple houses in the same instance… Surely, you see how demanding to have coherent P(Red xor Blue) becomes unfeasible?
The problem is, our intuitions are too eager to assume that everything as independent. We are used to think in terms of physical time, using our memory as something that allows us to orient in it. This is why amnesia scenarios are so mindboggling to us!
And that’s why the notion of probability experiment where every single trial is independent and the outcomes in any single trial are mutually exclusive is so important. We strictly define what the “situation” means and therefore do not allow ourselves to be tricked. We can clearly see that individual awakenings can’t be treated as outcomes of the Sleeping Beauty experiment.
But when you are thinking in terms of “reference classes” your definition of “situation” is too vague. And so you allow yourself to count the same house multiple times. Treat yourself not as a person participating in the experiment but as an “awakening state of the person”, even though one awakening state necessary follows the other.
The “point of probability” is lost when it doesn’t allign with reasoning about instances of probability experiments. Namely, we are starting to talk about something else, instead of what was logically pinpointed as probability in the first place. Most of the time reasoning about reference classes does allign with it, so you do not notice the difference. But once in a while it doesn’t and so you end up having “probability” that contradicts conservation of expected evidence and “utility” shifting back and forth.
So what’s the point of these reference classes? What’s so valuable in them? As far as I can see they do not bring anything to the table except extra confusion.
Upon rereading your posts, I retract disagreement on “mutually exclusive outcomes”. Instead...
An obvious way to do so is put a hazard sign on “probability” and just not use it, not putting resources into figuring out what “probability” SB should name, isn’t it? For instance, suppose Sleeping Beauty claims “my credence for Tails is 1π”; any specific objection would be based on what you expected to hear.
(And now I realize a possible point why you’re arguing to keep “probability” term for such scenarios well-defined; so that people in ~anthropic settings can tell you their probability estimates and you, being observer, could update on that information.)
As for why I believe probability theory to be useful in life despite the fact that sometimes different tools need to be used: I believe disappearing as a Boltzmann brain or simulated person is balanced out by appearing the same way, dissolving into different quantum branches is balanced out by branches reassembling, and likewise for most processes.
It’s an obvious thing to do when dealing with simularity clusters poorly defined in natural language. Not so much, when we are talking about a logically pinpointed mathematical concept which we know are crucial for epistemology.
It’s not just about anthropic scenarios and not just about me being able to understand other people. It’s about general truth preserving mechanism of logical and mathematical reasoning. When people just use different definitions—this is annoying but fine. But when they use different definitions without realizing that these definitions are different and, moreover insist that it’s you who is making a mistake—then we have an actual disagreement about math which will provide more confusion along the way. Anthropic scenarious are just the ones where this confusion is noticeable.
What exactly do you mean by “different tools need to be used”? Can you give me an example?
I mean that Beauty should maintain full model of experiment, and use decision theory as well as probability theory (if latter is even useful, which it admittedly seems to be). If she didn’t keep track of full setup but only “a fair coin was flipped, so the odds are 1:1”, she would predictably lose when betting on the coin outcome.
Also, I’ve minted another “paradox” version. I can predict you’ll take issue with one of formulations in it, but what do you think about it?
I suppose the participant is just supposed to lie about their credence here in order to “win”.
On Tuesday your credence in Heads supposed to be 0, but saying the true value would go against the experimental protocol unless you also said that your credence is 0 on Monday, which would also be a lie.
I don’t understand this formulation. If Beauty always says that the probability of Heads is 1⁄7, does she win? Whatever “win” means...
She certainly gets a reward for following experimental protocol, but beyond that… I concur there’s the problem, and I have the same issue with standard formulation asking for probability.
In particular, pushing problem out to morality “what should Sleeping Beauty answer so that she doesn’t feel as if she’s lying” doesn’t solve anything either; rather, it feels like asking question “is continuum hypothesis true?” providing only options ‘true’ and ‘false’, while it’s actually independent of ZFC axioms (claims of it or of its negation produce different models, neither proven to self-contradict).
P.S. One more analogue: there’s a field, and some people (experimenters) are asking whether it rained recently with clear intent to walk through if it didn’t; you know it didn’t rain but there are mines all over the field.
I argue you should mention the mines first (“that probability—which by the way will be 1⁄2 - can be found out, conforms to epistemology, but isn’t directly usable anywhere”) before saying if there was rain.
If you can demonstrate how, in the reference class setting, there is a relevant criterion by which several instances should be grouped together, then I think you could have an argument.
If you look at space-time from above, there’s two blue houses for every red house. Sorry I meant there’s two SB(=Sleeping Beauty)-tails instances for every SB-heads instance. The two instances you want to group together (tails-Monday & tails-Tuesday) aren’t actually at the same time (not that I think it matters). If the universe is very large of Many Worlds is true, then there are in fact many instances of Monday-heads, Monday-tails, and Tuesday tails occurring at the same time, and I don’t think you want to group those together.
In any case, from the PoV of SB, all instances look identical to you. So by what criterion should we group some of them together? That’s the thing I think your position requires (just because you accept reference classes are a priori valid and then become invalid in some cases), and I don’t see the criterion.