Radford Neal

Karma: 84

Radford Neal 20 Mar 2019 19:47 UTC
2 points
in reply to: dadadarren’s comment on: Sleeping Beauty Problem, Anthropic Principle and Perspective Reasoning
Sleeping Beauty with cookies is an almost-realistic situation. I could easily create an analogous situation that is fully realistic (e.g., by modifying my Sailor’s Child problem). Beauty will decide somehow whether or not to eat a cookie. If Beauty has no rational basis for making her decision, then I think she has no rational basis for making any decision. Denial of the existence of rationality is of course a possible position to take, but it’s a position that by its nature is one that it is not profitable to try to discuss rationally.

Radford Neal 18 Mar 2019 15:21 UTC
2 points
in reply to: dadadarren’s comment on: Sleeping Beauty Problem, Anthropic Principle and Perspective Reasoning
Regarding cloning, we have very good reason to think that good-enough memory erasure is possible, because this sort of thing happens in reality—we do forget things, and we forget all events after some traumas. Moreover, there are plausible paths to creating a suitable drug. For example, it could be that newly-created memories in the hippocampus are stored in molecular structures that do not have various side-chains that accumulate with time, so a drug that just destroyed the molecules without these side-chains would erase recent memories, but not older ones. Such a drug could very well exist even if consciousness has a quantum aspect to it that would rule out duplication.
I don’t see how your argument that the first person “me” perspective renders probability statements “invalid” can apply to Sleeping Beauty problems without also rendering invalid all uses of probability in practice. When deciding whether or not to undergo some medical procedure, for example, all the information I have about its safety and efficacy is of “third-person” form. So it doesn’t apply to me? That can’t be right.
It also can’t be right that Beauty has “no rational way of deciding to eat the cookie or not”. The experiment iis only slightly fantastical, requiring a memory erasing drug that could well be invented tomorrow, without that being surprising. If your theory of rationality can’t handle this situation, there is something seriously wrong with it.

Radford Neal 17 Mar 2019 2:23 UTC
2 points
in reply to: dadadarren’s comment on: Sleeping Beauty Problem, Anthropic Principle and Perspective Reasoning
I don’t think the issue of whether “cloning” is possible is actually crucial for this discussion, but since this relates to a common lesswrong sort of assumption, I’ll elaborate on it. I do think that making a sufficiently accurate copy is probably possible in principle (but obviously not now, and perhaps never, in practice). However, I don’t think this has been established. It seems conceivable that quantum effects are crucial to consciousness—certainly physics gives us no fundamental reason to rule this out. If this is true, then “cloning” (not the usual use of the word) by measuring the state of someone’s body and then constructing a duplicate will not work—the measurement will not be adequate to produce a good copy This possibility is compatible with there being some very good memory-erasing drug, which need only act on the quantum state of the person in a suitable way, without “measuring” it in its entirety. So I don’t agree with your statement that “if sleeping beauty problem is physically possible the cloning example must be as well”. And even if true in principle, there is a vast difference in practice between developing a slightly better amnesia drug—I wouldn’t be surprised if this was done tomorrow—and developing a way of measuring the state of someone’s brain accurately enough to produce a replica, and then also developing a way of constructing such a replica from the measurement data - my best guess is that this will never be possible.
This practical difference relates to a different sense in which your cloning example is “fantastic”. Even if we were sure that it was possible principle to “clone” people, we should not be sure that the methods of reasoning that we have evolved (biologically and culturally) will be adequate in a situation where this sort of thing happens. It would be like asking a human from 100,000 years ago to speculate on the social effects of Twitter. With social experience confined to a tribe of a dozen closely-related people, with occasional interactions with a few other tribes, not to mention a total ignorance of the relevant technology, they would be utterly incompetent to reason about how billions of people will interact when reacting to online text and video postings.
In this discussion, I get the impression that considering fantastical things like cloning leads you to discard common-sense realism. Uncrictially apply our current common-sense notions might indeed be invalid in a world where you can be duplicated—with the duplicate having perhaps first had its memories maliciously edited. There are lots of interesting, and difficult, issues here. But these are not issues that need to be settled in order to settle the Sleeping Beauty problem!
In your cloning example, you abandon common sense realism for no good reason. Since you talk about an original versus the clone, I take it that you see the experimenters as measuring the state of the original, without substantially disturbing it, and then creating a copy (as opposed to using a destructive measurement process, and then creating two copies, since then there is obviously no “original” left). In this situation, the distinction between the original and the copy is completely clear to any observer of the process. When they wake up, both the copy and the original do not know whether or not they are the original, but nevertheless one is the original and one is not. They can find out simply by asking the observer (and of course there are other possible ways—as is true for any fact about the world). Before they find out, they can assess the probability that they are the original, if that amuses them, or is necessary for some purpose. Nothing about this situation justifies abandoning the usual idea that probabilities regarding facts about the world are meaningful.
Regarding the cookies, you say “there is no strategy to maximize a “beauty in the moments” overall pleasure”. So once again, I ask: How is Beauty supposed to decide whether to eat a cookie or not?

Radford Neal 13 Mar 2019 14:35 UTC
2 points
in reply to: dadadarren’s comment on: Sleeping Beauty Problem, Anthropic Principle and Perspective Reasoning
“The probability of me being a man” in the anthropic sense means the probability of me being born into this world as a human male. Or it can been seen as the probability of my soul getting embodied as a human male. … even though “I’m a man” is a valid statement “the probability of me being a man” does not exist

Here, you have imported some highly questionable ideas, which would seem to be not at all necessary for analysing the Sleeping Beauty problem. This is my core objection to how Sleeping Beauty is used—it’s an almost-non-fantastical problem that people take to have implications for these sorts of anthropic arguments, but when correctly analysed, it does not actually say anthing about such issues, except in a negative sense of revealing some arguments to be invalid.

You should also note that your use of “probability” here does not correspond to any use of this word in normal reasoning. To see this, consider “the probability of my having blue eyes”. It take this to be in the same class as “the probability of me being a man”, but it allows for less-ridiculous thought experiments. Suppose you are a member of an isolated desert tribe. There are no mirrors, and no pools of water in which you could see your reflection. The tribe also has a strong taboo against telling anyone what colour their eyes are. So you don’t know what colour your eyes are. Do you maintain that “the probability that my eyes are blue” does not exist? Can’t you look at the other members of the tribe, see what fraction have blue eyes, and take that as the probability that you have blue eyes? Note that this may have practical implications regarding how much care you should take to avoid sun exposure, to reduce your chance of developing glaucoma.

I assume that you do think “the probability that my eyes are blue” is meaningful in this scenario. You seem to have in mind only something like prior probabilties, not conditional on any observations. But all actual practical uses of probability are conditional on observations, so your discussion is reminescent of the proverbial question of “how many angels can dance on the head of a pin?”.

I also agree maximizing someone’s own earning would force the decisions to reflect the probability.

I’m not sure what exactly you’re agreeing about here. Do you maintain that “the probability that it is Monday” does not exist, until Beauty happens to remember the other experiment, at which point it suddenly becomes meaningful? If so, why can’t Beauty just magine that there is some such practical reason to want to know whether it is Monday, calculate what the probability is, and then take that to be the probability of it being Monday even though she doesn’t actually need to make a decision for which that probability would be needed? Seems better than claiming that the probability doesn’t exist, even though this procedure gives it a well-defined value...

The whole reason sleeping beauty problem is related to anthropic reasoning is because it involves an observer duplication. … So they should have distinct rewards. Monday beauty’s correct decision should benefit Monday Beauty alone...

There’s a methodological issue here. I’ve presented a variation on Sleeping Beauty that I claim shows that “the probability that it’s Monday” has to be a meaningful concept for Beauty. You say, “but if I look at a different variation, that arguement doesn’t go through.” Why should that matter, though? If my variation shows that the probability is meaningful, that should be enough. If this shows that Sleeping Beauty is not related to anthropic reasoning, so be it.

However, there’s no problem making the reward be for “Beauty in the moment”. Suppose that when Beauty wakes up, she sees a plate of cookies. She recognizes them as being freshly baked by a bakery she knows. She also knows that on Mondays, but not Tuesdays, they put an ingredient in the cookies to which she is mildly allergic, causing immediate, painful stomach cramps. She also knows that the cookies are quite delicious. Should she eat a cookie? Adjust the magnitudes of possible pleasure and pain as desired to make the question interesting. Shouldn’t the probability of it being Monday be meaningful?

Suppose when you go to sleep tonight a clone of you would be created and put into an identical room. The clone is highly accurate it retains the memory good enough so he fully believes he is the one fall asleep yesterday.

Note that this is now a completely fantastical thought experiment, in contrast to the usual Sleeping Beauty problem. It may be impossible in principle, given the quantum no-cloning theorem. I also don’t know how this is supposed to work in conjunction with your previous reference to “souls”. I don’t think this extreme variation actually shows anything interesting, but if it did, you’d need to ask yourself whether the need to resort to this fantasy indicates that you’re in “angels dancing on the head of a pin” territory.

Radford Neal 11 Mar 2019 17:33 UTC
3 points
in reply to: dadadarren’s comment on: Sleeping Beauty Problem, Anthropic Principle and Perspective Reasoning
I’m afraid this makes no sense to me. I think this comes from my not understanding how the concept of a “reference class” can possible work. So I have no idea what it could mean to “observe the world from the perspective of any human that is male”, if observing from that “perspective” is supposed to change the probability (or render the probability meaningless) of some statement that I would take to be about the actual, real, world.
As I’ve pointed out before, the Sleeping Beauty problem is only barely a thought experiment—with a slight improvement over current memory-affecting drugs, it would be possible to actually run the experiment. It’s not like a thought experiment involving hypothetical computer simulation of people’s brains, or some such, in which one might perhaps think that common sense reasoning is not applicable.
So consider an actual run of the experiment. Suppose that at the time Beauty agrees to take part in the experiment, she fails to remember that she had already agreed to participate in a different experiment on Monday afternoon. The Sleeping Beauty experimenters have promised to pay her $120 if she completes their experiment, while the other experimenters have promised to pay her $120+X, and her motivation is to maximize the expected amount of her earnings. On some awakening during the Sleeping Beauty experiment, Beauty realizes that she had forgotten about the other experiment, and considers leaving to go participate in it. Of course, she then wouldn’t get the $120 for participating in the Sleeping Beauty experiment, but if it’s Monday, she would get the $120+X for participating in the other experiment. Now if it’s Tuesday, the other experiment has already been cancelled. So she needs to consider the probability that it’s Monday in order to make a good decision.
It’s not actually relevant to my point, but here is how it seems to me the probabilities work out. Suppose that Beauty has probability p of remembering the other experiment whenever she awakens, and suppose that this is independent for two awakenings (as is consistent with the assumption that her mental state is reset before a second awakening). To simplify matters, let’s suppose (and suppose that Beauty also supposes) that p is quite small, so the probability of Beauty remembering the other experiment on both awakenings (if two happen) is negligible.
Since p is small, Beauty’s probability for it being Monday given that she has woken and remembered the other experiment should be essentially as usual for this problem, with the answer depending on whether she is a Halfer or a Thirder. (If p were not small, she might need to downgrade the probability of Tuesday because there might be a non-negligible chance that she would have left the experiment on Monday, eliminating the Tuesday awakening.)
If she’s a Thirder, when she wakes and remembers the other experiment, she will consider the probability that it is Monday to be ²⁄₃, and will leave for the other experiment if (2/3)(120+X) is greater than 120, that is, if X is greater than 60. If she is a Halfer, it’s harder to say, since Halferism is wrong, but let’s suppose that she splits the ¹⁄₂ probability of two awakenings equally, and hence thinks the probability of it being Monday is ³⁄₄. She will then leave if (3/4)(120+X) is greater than 120, that is, if X is greater than 40. We can also look at things from a frequentist perspective, and ask what her expected payment is if she always decides to leave when she remembers the other experiment. It will be (1-p)120 + p(120+X) conditional on the coin landing Heads, and (1-p)(1-p)120 + p(120+X) conditional on the coin landing Tails, for a total expectation of (1-(3/2)p)120 + p(120+X), ignoring the p-squared term as being negligible. This simplifies to (1-p/2)120 + pX, which is greater than 120 if X is greater than 60, in agreement with the Thirder reasoning.
In any case, though, if I’ve understood you correctly, you deny that there is any meaning to “the probability that it’s Monday” in this situation. So how is Beauty supposed to decide what to do?

Radford Neal 10 Mar 2019 18:10 UTC
7 points
on: Sleeping Beauty Problem, Anthropic Principle and Perspective Reasoning
This is because in the context of sleeping beauty problem the probabilities of “today being Monday/Tuesday” do not exist. In another word “what’s the probability of today being Monday/Tuesday” are invalid questions.
I think here you depart from common-sense realism, in favour of what I am not sure.
From a common-sense standpoint, it is meaningful for Beauty to consider the probability of it being Monday because she can decide to batter down the door to her room, go outside, and ask someone she encounters what day of the week it is. That she actually has no desire to do this does not render the question meaningless.

Radford Neal 15 Aug 2018 16:29 UTC
1 point
in reply to: dadadarren’s comment on: The Sleeping Beauty Problem and The Doomsday Argument Can Be Explained by Perspective Inconsistency
What I mean by “someone with those memories exists” is just that there exists a being who has those memories, not that I in particular have those memories. That’s the “non-indexical” part of FNC. Of course, in ordinary life, as ordinarily thought of, there’s no real difference, since no one but me has those memories.
I agree that one could imagine conditioning on the additional piece of “information” that it’s me that has those memories, if one can actually make sense of what that means. But one of the points of my FNC paper is that this additional step is not necessary for any ordinary reasoning task, so to say it’s necessary for something like evaluating cosmological theories is rather speculative. (In contrast, some people seem to think that SSA is just a simple extension of the need to account for sampling bias when reasoning about ordinary situations, which I think is not correct.)

Radford Neal 13 Aug 2018 20:57 UTC
1 point
in reply to: dadadarren’s comment on: The Sleeping Beauty Problem and The Doomsday Argument Can Be Explained by Perspective Inconsistency
I can sort of see what you’re getting at here, but to me needing to ask “what question was being asked?” in order to do a correct analysis is really a special case of the need to condition on all information. When we know “the older child in that family is a boy”, we shouldn’t condition on just that fact when we actually know more, such as “I asked a neighbour whether the older child is a boy or girl, and they said ‘a boy’”, or “I encountered a boy in the family and asked if they were the older one, and they said ‘yes’”. Both these more detailed descriptions of what happened imply (assuming truthfulness) that the older child is a boy, but they contain more information than that statement alone, so it is necessary to condition on that information too.
For Technicolor Beauty, the statement (from Beauty’s perspective) “I woke up and saw a blue piece of paper” is not the complete description. She actually knows sometime like “I woke up, felt a bit hungry, with an itch in my toe, opened my eyes, and saw a fly crawling down the wall over a blue piece of paper, which fluttered at bit because the air conditioning was running, and I remembered that the air duct is above that place, though I can’t see it behind the light fixture that I can see there, etc.”. I argue that she should then condition on the fact that somebody has those perceptions and memories, which can be seen as a third-person perspective fact, though in ordinary life (not strange thought experiments involving AIs, or vast cosmological theories) this is equivalent to a first-person perspective fact. So one doesn’t get different answers from different perspectives, and one needn’t somehow justify disagreeing with a friend’s beliefs, despite having identical information.

Radford Neal 10 Aug 2018 12:56 UTC
1 point
in reply to: dadadarren’s comment on: The Sleeping Beauty Problem and The Doomsday Argument Can Be Explained by Perspective Inconsistency
I’m not sure what you’re saying in this reply. I read your original post as using the island problem to try to demonstrate that there are situations in which using probabilities conditional on all the available information gives the wrong answer—that to get the right answer, you must instead ignore “ad hoc” information (though how you think you can tell which information is “ad hoc” isn’t clear to me). My reply was pointing out that this example is not correct—that if you do the analysis correctly, you do get the right answer when you use all the information. Hence your island problem does not provide a reason not to use FNC, or to dismiss the Technicolor Beauty argument.
In the Technicolor Beauty variation, the red and blue pieces of paper on the wall aren’t really necessary. Without any deliberate intervention, there will just naturally be numerous details of Beauty’s perceptions (both of the external world and of her internal thoughts and feeling) which will distinguish the days. Beauty should of course reason correctly given all this information, but I don’t see that there are any subtle aspects to “how” she obtains the information. She looks at the wall and sees a blue piece of paper. I assume show knows that the experimenter puts a red or blue piece of paper on the wall. What is supposed to be the issue that would make straightforward reasoning from this observation invalid?

Radford Neal 8 Aug 2018 17:38 UTC
2 points
on: The Sleeping Beauty Problem and The Doomsday Argument Can Be Explained by Perspective Inconsistency
As you may know, my Full Nonindexical Conditioning (FNC) approach (see http://www.cs.utoronto.ca/~radford/anth.abstract.html) uses the third-person perspective for all inference, while emphasizing the principle that all available information should be used when doing inference. In everyday problems, a third-person approach is not distinguishable from a first-person approach, since we all have an enormous amount of perceptions, both internal and external, that are with very, very high probability not the same as those of any other person. This approach leads one to dismiss the Doomsday Argument as invalid, and to adopt the Thirder position for Sleeping Beauty.
You argue against approaches like FNC by denying that one should always condition on all available information. You give an example purporting to show that doing so is sometimes wrong. But your example is simply mistaken—you make an error somewhat analogous to that made by many people in the Monte Hall problem.
Here is your example (with paragraph breaks added for clarity):
Imagine you are on an exotic island where all families have two children. The island is having their traditional festival of boys’ day. On this day it is their custom for each family with a boy to raise a flag next to their door. Tradition also dictates in case someone knocks on the door then only a boy can answer. You notice about ³⁄₄ of the families has raised a flag as expected. It should be obvious that if you randomly choose a family with flags then the probability of that family having two boys is ¹⁄₃. You also know if
you knock on the door a boy would come to answer it so by seeing him there is no new information.
But not so fast. When you see the boy you can ask him “are you the older or the younger child?”. Say he is the older one. Then it can be stated that the older child of the family is a boy. This is new information since I could not know that just by seeing the flag. If both children are boys then the conditional probability of the older kid being a boy is one. If only one child is a boy then the conditional probability of the older kid being a boy is only half. Therefore this new evidence favours the former. As a result the probability of this family having 2 boys can be calculated by bayesian updating to increase to be ¹⁄₂. If the child is the younger kid the same reason can still be applied due to symmetry. Therefore even before knocking on the door I should conclude the randomly chosen family’s probability of having 2 boys is ¹⁄₂ instead of ¹⁄₃.
This is absurd. This shows specifying the child base on ad hoc details is clearly wrong. For the same reason I should not specify today or this awakening by ad hoc details observed after waking up, such as the color of the paper.
Your mistake here is in asking the boy “are you the older or younger child?” and then reasoning as if an “older” answer to this question is the same as a “yes” answer to the question “is the older child in this family a boy?”.
If you actually ask a neighbor “is the older child in that family a boy?”, and get the answer “yes”, then it WOULD be correct to infer that the probability of the younger child also being a boy is ¹⁄₂. But you didn’t do that, or anything equivalent to that, as can be seen from the fact that the question you actually asked cannot possibly tell you that the older child is a girl.
The correct analysis is as follows. Before knowing anything about the family, there are four equally likely possibilities, which we can write as BB, BG, GB, GG, where the first B or G is the sex of the younger child, and the second is the sex of the older child. When you see the flag on the family’s house, the GG possibility is eliminated, leaving BB, BG, GB, all having probability ¹⁄₃. When a boy answers the door, the probabilities stay the same. After you ask whether the boy is the younger or older child, and get the answer “older”, the likelihood function over these three possibilities is ¹⁄₂, 0, 1, which when multiplied by ¹⁄₃, ¹⁄₃, ¹⁄₃ and renormalized gives probability ¹⁄₃ to BB and probability ²⁄₃ to GB, with zero probability for BG (and GG). If instead the answer is “younger”, the result is probability ¹⁄₃ for BB and ²⁄₃ for BG.
There is nothing odd or absurd here. Conditioning on all available information is always the right thing to do (though one can ignore information if one knows that conditioning on it won’t change the answer).

Radford Neal 13 Jul 2018 0:23 UTC
1 point
in reply to: Said Achmiz’s comment on: Probability is a model, frequency is an observation: Why both halfers and thirders are correct in the Sleeping Beauty problem.
But the thing is you can’t call it “0.5 credence” and have your credence be anything like a normal probability. The Halfer will assign probability ¹⁄₂ for Heads and Monday, ¹⁄₄ for Tails and Monday, and ¹⁄₄ for Tails and Tuesday. Since only the guess on Monday is relevant to the payoff, we can ignore the Tuesday possibility (in which the action taken has no effect on the payoff), and see that a halfer would have a 2:1 preference for Heads. In contrast, a Thirder would give ¹⁄₃ probability to Heads and Monday, ¹⁄₃ to Tails and Monday, and ¹⁄₃ to Tails and Tuesday. Ignoring Tuesday, they’re indifferent between guessing Heads or Tails.
With a slight tweak to payoffs so that Tails are slightly more rewarding, the Halfer will make a definitely wrong decision, while the Thirder will make the right decision.

Radford Neal 13 Jul 2018 0:16 UTC
1 point
in reply to: Said Achmiz’s comment on: Probability is a model, frequency is an observation: Why both halfers and thirders are correct in the Sleeping Beauty problem.
Well of course. If we know the right action from other reasoning, then the correct probabilities better lead us to the same action. That was my point about working backwards from actions to see what the correct probabilities are. One of the nice features about probabilities in “normal” situations is that the probabilities do not depend on the reward structure. Instead we have a decision theory that takes the reward structure and probabilities as input and produces actions. It would be nice if the same nice property held in SB-type problems, and so far it seems to me that it does.
I don’t think there has ever been much dispute about the right actions for Beauty to take in the SB problem (i.e., everyone agrees about the right bets for Beauty to make, for whatever payoff structure is defined). So if just getting the right answer for the actions was the goal, SB would never have been considered of much interest.

Radford Neal 12 Jul 2018 23:43 UTC
1 point
in reply to: Said Achmiz’s comment on: Probability is a model, frequency is an observation: Why both halfers and thirders are correct in the Sleeping Beauty problem.
A big reason why probability (and belief in general) is useful is that it separates our observations of the world from our decisions. Rather than somehow relating every observation to every decision we might sometime need to make, we instead relate observations to our beliefs, and then use our beliefs when deciding on actions. That’s the cognitive architecture that evolution has selected for (excepting some more ancient reflexes), and it seems like a good one.

Radford Neal 12 Jul 2018 23:34 UTC
3 points
in reply to: Said Achmiz’s comment on: Probability is a model, frequency is an observation: Why both halfers and thirders are correct in the Sleeping Beauty problem.
The linked post by ata is simply wrong. It presents the scenario where
Each interview consists of Sleeping Beauty guessing whether the coin came up heads or tails. After the experiment, she will be given a dollar if she was correct on Monday.
In this case, she should clearly be indifferent (which you can call “.5 credence” if you’d like, but it seems a bit unnecessary).
But this is not correct. If you work out the result with standard decision theory, you get indifference between guessing Heads or Tails only if Beauty’s subjective probability of Heads is ¹⁄₃, not ¹⁄₂.
You are of course right that anyone can just decide to act, without thinking about probabilities, or decision theory, or moral philosophy, or anything else. But probability and decision theory have proven to be useful in numerous applications, and the Sleeping Beauty problem is about probability, presumably with the goal of clarifying how probability works, so that we can use it in practice with even more confidence. Saying that she could just make a decision without considering probabilities rather misses the point.

Radford Neal 12 Jul 2018 23:18 UTC
2 points
in reply to: shminux’s comment on: Probability is a model, frequency is an observation: Why both halfers and thirders are correct in the Sleeping Beauty problem.
The standard question is what probability Beauty should assign to Heads after being woken (on Monday or Tuesday), and not being told what day it is, given that she knows all about the experimental setup. Of course if you change the setup so that she’s asked a question on Monday that she isn’t on Tuesday, then she will know what day it is (by whether the question was asked or not) and the answer changes. That isn’t an interesting sense in which the answer ¹⁄₂ is correct. Neither is it interesting that ¹⁄₂ is the answer to the question of what probability the person flipping the coin should assigns to Heads, nor to the question of what is seven divided by two minus three...

Radford Neal 12 Jul 2018 23:09 UTC
1 point
in reply to: Said Achmiz’s comment on: Probability is a model, frequency is an observation: Why both halfers and thirders are correct in the Sleeping Beauty problem.
We agree about what the right actions are for the various reward structures. We can then try to work backwards from what the right action is to what probability Beauty should assign to the coin landing Heads after being wakened, in order that this probability will lead (by standard decision theory) to her taking the action we’ve decided is the correct one.
For your second scenario, Beauty really has to commit to what to do before the experiment, which means this scheme of working backwards from correct decision to probability of Heads after wakening doesn’t seem to work. Guessing either Heads or Tails is equally good, but only if done consistently. Deciding after each wakening without having thought about it beforehand doesn’t work well, since with the two possibilities being equally good, Beauty might choose differently on Monday and Tuesday, with bad results. Now, if the problem is tweaked with slightly different rewards for guessing Heads correctly than Tails correctly, we can avoid the situation of both guesses being equally good. But the coordination problem still seems to confuse the issue of how to work backwards to the appropriate probabilities (for me at least).
I think it ought to be the case that, regardless of the reward structure, if you work backwards from correct action to probabilities, you get that Beauty after wakening should give probability ¹⁄₃ to Heads. That seems to be what happens for all the reward structures where Beauty can decide what to do each day without having to know what she might do or have done the other day.

Radford Neal 12 Jul 2018 21:36 UTC
1 point
in reply to: Said Achmiz’s comment on: Probability is a model, frequency is an observation: Why both halfers and thirders are correct in the Sleeping Beauty problem.
Your second scenario introduces a coordination issue, since Beauty gets nothing if she guesses differently on Monday and Tuesday. I’m still thinking about that.
If you eliminate that issue by saying that only Monday guesses count, or that only the last guess counts, you’ll find that Beuaty has to assign probability ¹⁄₃ to Heads in order to do the right thing by using standard decision theory. The details are in my comment on the post at https://www.lesswrong.com/posts/u7kSTyiWFHxDXrmQT/sleeping-beauty-resolved#aG739iiBci9bChh5D
Or you can say that the payoff for guessing Tails correctly is $0.50 while guessing Heads correctly gives $1.00, so the total payoff is the same from always guessing Heads as from always guessing Tails. In that case, you can see that you get indifference to Heads versus Taills when the probability of Heads is ¹⁄₃, by computing the expected return for guessing Heads at one particular time as (1/3) 1.00 versus the expected return for guessing Tails at one particular time of (2/3) 0.5. Clearly you don’t get indifference if Beauty thinks the probability of Heads is ¹⁄₂.

Radford Neal 12 Jul 2018 17:30 UTC
2 points
on: Probability is a model, frequency is an observation: Why both halfers and thirders are correct in the Sleeping Beauty problem.
Probability is meant to be a useful mental construct, that helps in making good decisions. There’s a standard framework for doing this. If you apply it, you find that Beauty makes good decisions only if she assigns a probability of ¹⁄₃ to Heads when she is woken. There is no sense in which ¹⁄₂ is the correct answer, unless you choose to redefine what probabilities mean, along with the method of using the to make decisions, which would be nothing but a pointless semantic distraction.

Radford Neal 9 Jul 2018 1:12 UTC
−1 points
in reply to: Rob Bensinger’s comment on: The Fermi Paradox: What did Sandberg, Drexler and Ord Really Dissolve?
I agree that the word “Paradox” was some sort of hype. But I don’t think anyone believed it. Nobody plugged their best guesses for all the factors in the Drake equation, got the result that there should be millions of advanced alien races in the galaxy, of which we see no sign, and then said “Oh my God! Science, math, and even logic itself are broken!” No. They instead all started putting forward their theories as to which factor was actually much smaller than their initial best guess.

Radford Neal 28 Jun 2018 13:13 UTC
2 points
in reply to: Chris_Leong’s comment on: The Beauty and the Prince
I’ve noticed something that may explain some of the confusion. You say above:
...halvers don’t believe that you can answer: “If Sleeping Beauty is awake, what are the chance that the coin came up heads?” without de-indexicalising the situation first.
But in the Sleeping Beauty problem as usually specified, the question is what probability Beauty should assign to Heads, not what some external observer should think she should be doing. Beauty is in no doubt about who she is (eg, she’s the person who just stubbed her toe on this bedpost here) even though she doesn’t know what day of the week it is.