A Simplified Version of Perspective Solution to the Sleeping Beauty Problem

by dadadarren5 min read31st Dec 20209 comments

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AnthropicsSleeping Beauty ParadoxWorld Modeling
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This post attempts to give a more intuitive explanation of my perspective-based solution to anthropic paradoxes. By using examples, I want to show how perspectives are axiomatic in reasoning. 

For each of us, our perceptions and experience originate from a particular perspective. Yet when reasoning we often remove ourselves as the first-person and present our logic in an objective manner. There are many names for this way of thinking, e.g. “take the third-person view”, “use an outsider’s perspective”, “think as an impartial observer”, “the god’s eye view”, “the view from nowhere”, etc.

I think the best way to highlight the differences between god’s-eye-view reasoning and first-person reasoning is to use examples. Imagine during Alex’s sleep some mad scientist placed him in a maze. After waking up, he sees the scientist left a map of the maze next to him. Nonetheless, he is still lost. Here we can say, given the map, the maze is known. Alex is lost because he doesn’t know his location.


 

Figure 1. From a god’s eye view, the maze’s location is known. But Alex’s location is unknown. The colored dots show some possibilities given Alex’s immediate surroundings.

In contrast, an alternative interpretation is to think in Alex’s shoes. I can say of course I know where I am. Right here. I can look down and see the place I am standing on. No location in the world is more clearly presented to me than here. I’m lost because I don’t know how is the maze located relative to me, even with the map.

Figure 2. Here is known. The maze’s location is unknown. The colored parts show some possibilities.

This difference also exists in examples involving time. Imagine a thunder wakes Alex up at night. Here we can say he doesn’t know what time it is. Maybe he went to bed at 10 pm and his alarm clock rings at 6 am. There is no information to place his awakening on the time axis.

Figure 3. From a god’s eye view, it is known when Alex went to bed and when the alarm clock is going to ring. But the time when the thunder wakes him up is not known. Colored lines show some possibilities.

Alternatively, we can think from Alex’s perspective as he wakes up. I can say of course I know what time it is. It is now. It is the time most immediate to my perception and experience. No other moment is more clearly felt. What I don’t know is how long ago was 10 pm when I went to bed, and how far in the future will my alarm clock ring at 6 am, i.e. how other moments locate relative to now.

Figure 4. Now is known. But how other moments locate relative to now is unknown. The colored part shows some possibilities.

The same difference also exists when identities are involved. Imagine Alex and Adam are identical twins. They got into a fatal accident while traveling in the same vehicle. One of them died while the other suffers complete memory loss. If they look the same and have the same belongings, it can be said there is no way to tell who survived the tragedy.

Figure 5. The past is known. The survivor is unknown. The colored parts show two possible outcomes.

There are other ways to describe the situation. For example, if we take the perspective of the survivor, it is obvious who survived the accident. I did. This self-identification is based on immediacy to the only subjective experience available: all senses felt are due to this body. No other persons or things are more closely perceived. Whether I was Alex or Adam, however, is unknown. In a sense, I don’t know how was the world located relative to me.

Figure 6. Obviously, I survived the accident while the other twin didn’t. However, the past is unknown. Whether I was Adam or Alex is lost. Colored parts show the two possibilities.

These examples show how different perspectives analyze problems differently. Without getting into all the details I want to stress the following points:

  1. First-person thinking is self-centered. Special consideration is given to here, now, and I. The perspective center is regarded as primitively understood due to its closeness to perception and subjective experience. It is the reasoning starting point. Other locations, moments, and identities are defined by their relative relations to the perspective center.
  2. First-person and god’s-eye-view are two distinct ways of reasoning. They should not be used together. For example, in the maze problem, we cannot say both the maze’s and Alex’s locations are known. While the former is true from a god’s-eye-view and the latter is true from Alex’s first-person perspective, mixing the two would make the whole “lost in the maze” situation unexplainable.

Anthropic problems are unique because they are formulated from specific first-person perspectives (or a set of perspectives). There is no straightforward god’s-eye-view alternative.

Take the sleeping beauty problem as an example. It asks for the probability when beauty wakes up in the experiment. However, there may be two awakenings. From a god’s eye view, the problem is not fully specified. Which awakening is being referred to? Why update the probability base on that particular awakening? But things are clear from beauty’s first-person perspective. I should obviously update my belief base on my experience: base on this awakening right now, the one I have experience of. From beauty’s point of view, today is primitively understood, no clarification is needed to differentiate it from the other day. And we intuitively recognize the problem is asking about a specific awakening/day.

Paradoxes ensue when we try to answer the problem from a god’s eye view while keeping that intuition. As pointed above, the awakening is not specified from god’s eye view.  We mistakenly try to fill in the blanks with additional assumptions. For example, regard this awakening as randomly selected from all awakenings (Self-Sampling Assumption), or regard today as randomly selected from all days (Self-Indication Assumption). By redefining the first-person center from the god’s eye view, these assumptions provide the seemingly missing part of the question: which specific awakening is being referred to, and why update the probability base on that particular awakening.

Different anthropic assumptions give different answers, which also leads to different paradoxes. However, these assumptions are redundant because the question should have been answered the same way it is asked: from Beauty’s first-person perspective. There is no reference class for I, here, and now. They are primitively identified right from the start, not selected from some similar group. Even the superficially innocent notion of “I am a typical observer” or “now is a typical moment” is false. From a first-person perspective,  I, now, and here are inherently special as shown by the self-centeredness. 

The correct answer is simple. From Beauty’s viewpoint, there is no new information as I wake up in the experiment. Because finding myself awake here and now is logically true from the first-person perspective. The probability stays at 1/2. From a god’s eye view, all that can be said here is there’s (at least) one awakening in the experiment. 

Some argue there is new information. That being awake on a specific day is evidence suggesting there are more awakenings. Halfers typically disagree by saying it is unknown which day it is. Thirders counter that by saying Beauty knows it is today. Halfers often argue today is not an “objective” identification. Thirders response by saying all the details Beauty sees after waking up can be used to identify the particular day. For example, the experimenter can randomly choose one day and paint the room red and paint the room blue on the other day. If Beauty sees red after waking up, then we know Beauty is awake on the Red day. Which is unknown before. Some argue this new information favors more awakenings. 

To that, I would ask: out of the two days, why update the probability base on what happens on the Red day? Why not focus on what happens on the Blue day and update the probability instead? Thirders may find this question painfully trivial: because what happens on the Blue day is not available. Beauty is experiencing the Red day. That’s reasonable. However, that means it doesn’t matter if we use the word today or use details observed after waking up, that particular day is still specified from Beauty’s first-person perspective. Yet from a first-person perspective finding myself awake here and now is expected.  The argument for new information switches to a god’s eye view. As if an outsider randomly selects the Red day then finds Beauty awake. This argument depends on first-person reasoning -to specify the day, and god’s-eye-view reasoning -to calculate the prior probability of finding Beauty awake on said day. It mixes parts from two perspectives. (For a more detailed analysis see my argument against Full Non-Indexical Conditioning and perspective disagreements)

As Beauty wakes up in the experiment, some may ask “what is the probability that today is Monday?”. The answer is interesting: there is no such probability. Remember as the first-person, the perspective center is the reasoning starting point. This includes I, here, now, and by extension today. Like axioms, they cannot be explained by logic or underlying mechanics. They are identified by intuition- by their apparent closeness to subjective experience. So there is no logical way to formulate a probability about what the perspective center is. 

One has to mix first-person and god’s-eye-view to validate this probability. For example, we can do so by accepting any one of the anthropic assumptions. They redefine today/this awakening as a randomly selected sample from some proposed reference class. Doing so means we can reason from a god’s-eye-view and justifies the principle of indifference to all days/awakenings. However, as discussed before, this mix is false. Now is a first-person concept that is inherently unique from all other moments. A principle of indifference categorically contradicts the self-centeredness of the first-person view. (For a more detailed discussion against self-locating probability, and why they are invalid from the frequentist or decision-making approach see my argument here).

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9 comments, sorted by Highlighting new comments since Today at 9:47 AM
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Being new to this, I have no problem asking naïve questions, so: why does Sleeping Beauty need to have any "credence" at all?  She's armed with the facts of the experiment and can make decisions based solely on those; why does anyone suppose that she forms some "credence" as a proxy for the facts?

Just like to add that I found your website very clear and, in parts, quite compelling.  Thank you.

I think your question is not naive at all. Stuart_Armstrong argued for a similar point that anthropic questions should be about decision making rather than probability assigning here. However I do remember a post in lesswrong claiming he has modified his position but I couldn't find that post right now.

That said, I think ignoring probability in anthropics is not the way to go. We generally regard probability as the logical basis of rational decision making. There is no good reason why anthropic problems are different. In my opinion, focusing on decision making mitigate one problem: it forces us to state the decision objective as premises. So Halfers and Thirders can each come up with their own objectives reflecting their answers. It avoids the question of which objective is reflective of the correct probability. By perspective-based reasoning, I think the correct objective should be simple selfish goals, where the self is primitively identified I by its immediacy to subjective experience (i.e. the perspective center). And for some paradoxes such as the doomsday argument and presumptuous philosopher, converting them into decision making problem seems rather strange: I just want to know if their conclusions are right or wrong, why even ask if I care about the welfare of all potentially existing humans?

Thank you for the quick reply.

I would agree that having a clear decision objective is important.  I would go further: without an objective, why should anyone care how the agent (who, by definition, takes action) feels about their circumstances?  I note in your final sentence that you see things differently, but I don't have a killer argument to the contrary.

I can see the need for subjective probability, but only in model selection.  Thereafter, you're working to find a strategy maximising the expected value of an objective function.  I recorded my thoughts here.

There is a simple way to answer the question without resorting anthropic reasoning. Then you can try to make the anthropic reasoning fit the (correct) answer.

Flip two coins on Sunday Night, a Dime and a Quarter. Lock them in a glass box showing the results. On Monday morning, perform this procedure: Look at the two coins. If either is showing Tails, wake SB, interview her, and put her back to sleep with the amnesia drug. If both are showing Heads, leave her asleep.

On Monday night, open the box, turn the Dime over, and re-lock it. On Tuesday morning, repeat the same procedure described for Monday.

In the interview(s), ask SB for her credence that the Quarter is currently showing Heads. Since the Quarter is never changed, it is always showing the same face that was the result of the flip. SB knows that when the locked box was examined, there are four equally probably possibilities for {Dime,Quarter}. They are HH, HT, TH, and TT. Since SB is now awake, she knows that HH is eliminated as a possibility. Her credence for each of TH, HT, and TT are each 1/3.

The only difference between this version, and the original, is that we don't need to say anything about what day it is.

I fail to see how this variation is going to settle the debate. Thirders will agree with your solution but halfers would disagree with it the same way as in the original sleeping beauty problem.

Halfers will ask why should beauty regard the four outcomes (HH, HT, TH, TT) equal probable? Yes they are equal probables if this is a simple tossing of two coins. Yet the experiment is far from that simple: my awakenings depend on it, the dime is being manipulated half way, my memory is erased after the first awakening....Halfers will say HH is never in the sample space to begin with, and there is no good reason to believe HT, TH and TT are equal probables. Beauty should just examine the information she has once waken up. She knew that she would definitely find herself awake in the experiment since the dime is manipulated, so right now being awake gives no new information about the Quarter, the probability ought to remain at 1/2. The same old dispute as in the original sleeping beauty.

Without trying to figure out the correct way to interpret today or this awakening the debate is not going to be settled. Some halfers (SSA) think this awakening shall be interpreted as a random awakening. Some thirders (SIA) think this awakening should be regarded as a random sample from all potential awakenings (thus being actually awake gives new information as the case of your argument). There are others (FNC) who think we should ignore indexicals such as today or this awakening all together but consider all objective information available. And I'm suggesting treating indexicals like fundamentals: they are primitively understood from the first-person perspective and irreducible. These are all attempts to solve the anthropic mystery. I don't think this debate can magically go away just by using a different experiment setup.

Let's say Beauty is paid (once, at the end) if & only if she guesses the Quarter correctly on every wake-up.  The reward will be £111 if she correctly guessed Heads, £100 if she correctly guessed Tails.  So she should guess Heads.

You would still say that her credence for Heads was , but you'd argue that she adapts her bet to take account of the experimental protocol, betting in defiance of her greater credence for Tails.  Right?

Now if the single non-HH Monday morning wake-up was replaced with some undisclosed number of wake-ups, I imagine you'd say that her credence was undefined.  Yet she's still able to take the bet, absent of any credence.  How is this?

The answer is that she does not need to resort to credence in making the decision.  So it is vacuous to argue that Halfing or Thirding is the "correct" approach.

Why does she get paid only once, at the end? Why not once for each waking?

This is the problem with all betting arguments. They incorporate an answer to the anthropic question by providing one, or #wakings, payoffs. 

She gets paid once because that's how I choose to demonstrate my point, supported by your reply, that arguments concerning the "correctness" of halving/thirding are impotent.

From a completely subjective view, there can be no single answer as to what day of the week it is. It is whatever weekday  I decide it is. Who could gainsay it?

As soon as we allow an objective view of what day of the week it is, we implicitly allow an over-ride of the subjective viewpoint by the objective one, and the 1/3 probability becomes the better choice.