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.
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.