The Sleeping Beauty Problem and The Doomsday Argument Can Be Explained by Perspective Inconsistency

post by dadadarren · 2018-08-05T13:45:27.185Z · LW · GW · 24 comments

This is a link post for https://www.sleepingbeautyproblem.com/perspective-consistency-and-the-solution-to-the-doomsday-argument-and-the-sleeping-beauty-problem/

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comment by jessicata (jessica.liu.taylor) · 2018-08-05T20:25:35.047Z · LW(p) · GW(p)
Same as before, the questions of “what is the probability that today is day 1?” or “what is the probability that this is the first awakening?” are invalid questions which have no answer. It is because these questions cannot be fully interpreted from either perspectives. From first-person perspective the question makes no sense because to me this is indeed the one and only awakening. The date or the sequence of the awakening is only meaningful from a third-person perspective.

This doesn't seem right. Say there is a calendar in the room next to Beauty's that indicates the current day. Then, from a first-person perspective, Beauty can refer to the events "the calendar in the room next to mine says it is Monday" or "the calendar in the room next to mine says it is Tuesday". Even without the calendar in the room next to hers, she could refer to "the world I am in at the current time" and thereby refer to random variables like "the day that the majority of calendars in the world I am in currently say it is". Uncertainty about these events and random variables is the same as ordinary physical uncertainty (e.g. being uncertain about what color marble is in a particular box).

Replies from: dadadarren
↑ comment by dadadarren · 2018-08-07T02:48:49.219Z · LW(p) · GW(p)

As explained earlier in the blog, from my first-person perspective whoever wakes up on the other day is a different person. So from my perspective the amount of time passed is not the same as everyone else's. That's why for me this is indeed the first day. To ask about the calendar in the other room is to switch to a third-person's (who have not experienced a memory wipe) perspective thus the perspective inconsistency. The event of "the calendar in the room next to mine says it is Monday" and "the calendar in the room next to mine says it is Tuesday" refers to two different persons and cannot be both in my sample space. Regarding its difference from an ordinary physical uncertainty I have discussed it in the chapter regarding the doomsday argument. The exact same reason applies here. To summarized it the probability cannot be interpreted in frequentist's sense because there is no experiment to repeat. Principle of indifference cannot be used either because defining "today" basing on immediacy to perception already violates it. Where as for a physical uncertainty both interpretation works.

If I may make a plea. I find discussing duplication by memory wipes very difficult because it is hard to put into words. IF you agree that duplication by cloning is logically the same problem can we proceed with that route?

Replies from: jessica.liu.taylor
↑ comment by jessicata (jessica.liu.taylor) · 2018-08-07T06:09:22.363Z · LW(p) · GW(p)

Agree that the cloning problem is a fine problem to discuss instead of sleeping beauty, and the analysis is mostly the same.

To ask about the calendar in the other room is to switch to a third-person’s (who have not experienced a memory wipe) perspective thus the perspective inconsistency.

Huh? Beauty might even be able to go into the other room and check the calendar. Everyone else in the world could be dead and she could still go check the calendar. I'm really confused about how talking about this calendar requires a different person's perspective; there are few things more immediate to Beauty's first-person experience than a calendar she can just go look at.

The event of “the calendar in the room next to mine says it is Monday” and “the calendar in the room next to mine says it is Tuesday” refers to two different persons and cannot be both in my sample space.

From a first-person perspective, it doesn't refer to any persons other than "me".

I think this gets less confusing if we think of an apparently non-anthropic problem. Say a friend has put either a red marble or a blue marble into a box (without showing me what color marble), and puts the box in front of me. Then I look at the box, and think, "I wonder what color marble is in the box". I can form two events: "the marble in the box in front of me is red" and "the marble in the box in front of me is blue", and be uncertain about which of these events is true. (These events make formal sense in centered world models)

Other people have boxes containing marbles in front of them, and could also form the events "the marble in the box in front of me is red" and "the marble in the box in front of me is blue". But there's no way for me to confuse myself with these other people, or my box with theirs; it's not like I would accidentally walk into someone else's room and look at their marble-containing box instead of mine. The box is quite immediate in my own experience, and I can refer to it without referring to any person who isn't me.

It would be incorrect to say that the event "the marble in the box in front of me is red" refers to many different persons and cannot be in my sample space.

I don't think I can use repeatable experiments or the principle of indifference to reason about the probability of the marble being blue or red in this non-anthropic experiment, yet I still assign a probability. There are reasons for having probabilities that don't rely on repeatability or principle of indifference (see: complete class theorem, Cox's theorem, dutch book arguments, Occam's razor). It remains an open problem how to assign probabilities in the absence of repeatibility or principle of indifference, but this is a real problem with real consequences rather than a meaningless question.

Replies from: dadadarren
↑ comment by dadadarren · 2018-08-10T03:48:00.332Z · LW(p) · GW(p)
Huh? Beauty might even be able to go into the other room and check the calendar. Everyone else in the world could be dead and she could still go check the calendar. I'm really confused about how talking about this calendar requires a different person's perspective; there are few things more immediate to Beauty's first-person experience than a calendar she can just go look at.

Here I don't mean that beauty has to check the calendar through another person. Think this way. If beauty has a calendar of her own which also goes though the "memory wipe" as she does then for sure it would say it's Monday. Beauty knows this as well. For the calendar to be indeterminate to beauty it has to be outside of the experiment unaffected by the memory tempering. If we can say the calendar is a valid perspective then to beauty it is an third-person perspective just as the experimenter's or an outside observers'. That's what I meant by checking the calendar to determined the date beauty is switching to a third-person perspective.

I personally do not like the version of the problem where the awakenings happen on two specific dates. For sure it makes some arguments easier to express but I feel it also give a false sense where only the outsider's perspective is valid because only they know the date. I think calendar in the context of this question is just a consensus of a referencing point to account the passing of time, nothing special to it. So beauty could rightfully have her own calendar different from everybody else. However I think these arguments are hard to convey and not really the point of contention anyway. That's why in my blog I did not specify the dates. I think we can all agree for the experiment the specific dates or the timing of the awakening doesn't really matter. Only the number of awakenings matters to the answer. So the real question is whether this awakening is the first or the second, (or alternatively if this is the awakening that would always happen or the one that only happens if tails). The concept of first vs second is only meaningful from the third-person perceptive of an outsider not affected by the memory wipe. But using "this awakening" or "today" to specify a time is purely based on immediacy to perception, i.e. first-person. Hence the perspective inconsistency, in another word, the question is formed by using part of the logic from one person and combine it with part of the logic from another.

If I may I want to use a cloning example instead of memory wiping to discuss the difference between these perspectively inconsistent problems and normal non-anthropic problems. Here I'm going to cut the corner and just use one of my example from the footnote. Consider a case where memory retained cloning has created two indistinguishable individuals and you are one of them. Each person is put into an identical room. Someone would randomly choose one out of the two rooms and paint it red and paint the other room blue. It is perfectly valid to ask "what is the probability that I would open my eyes and see a red room (instead of a blue one)?". I think even though this thought experiment involves anthropic reasoning it is analogous to the marble in box problem you presented. Here the clones are only specified in first-person. Whatever happens to others is not interested to the problem. It keeps a constant perspective and it's a valid question. The experiment refers to the random choosing of the rooms. To answer the question the experiment can be repeated to count the relative frequency. And the two outcomes of "red room" vs "blue room" to me are indifferent. A perspectively inconsistent question would be "what is the probability of me being the clone in the red room (instead of being the clone in the blue room)?" I know this expression is a bit weird. Here the individuals are defined in first-person as in "me", but also in third-person as the two clones in different rooms. To make sense of the question I have to combine the perspective of me with the perspective of some observer. That's whats make the question fallacious. There really is no experiment to the question. Except maybe I can make up a fictitious one where my wandering soul finds one of the clone to get embodied. Then the sample space would be for one person (me) to become two possible persons (the red clone or the blue clone). Even if we use this made up experiment there is no way of repeating it to get a frequentist interpretation. Also because part of the question is from first-person perspective it means not all clones are treated equally (since it treats the first-person as an inherently special individual). It would be self contradicting to treat each clone indifferently in its answer. For example, a common argument would be if all clones guessed "red" then half would be correct so my probability of being the red clone is 1/2. This argument only works if the "I" in question is specified in third-person. If I was defined in first-person then I'm already inherently unique and has no one else in my reference class. Basing my answer on averaging me with others makes no sense at all. What I want to stress is that even though the two question looks similar the latter is actually invalid and has no answer. But because the questions are similar we usually just treat the latter as an alternative expression of the former without realizing that. However for questions like "what is the probability of me being the original (instead of being the clone)?" or in the sleeping beauty problem "what is the probability of this awakening being the first (instead of being the second)?" There is no readily available former question. SIA and SSA are just attempts trying to come up with one. In my opinion these are ultimately pointless.

PS: Sorry for the slow reply. Had to take a 14 hour flight with my one year old. Reading your questions made me realise how poorly I expressed myself in the first reply. Really hoping you are still interested in this discussion. I find it really challenging and at the same time forcing me to try to articulate my argument.

Replies from: jessica.liu.taylor
↑ comment by jessicata (jessica.liu.taylor) · 2018-08-10T04:14:14.167Z · LW(p) · GW(p)

I wasn't thinking of the calendar as a perspective, rather as a particular material object, which could even be a mechanical clock. (Events like "the clock reads Monday" are material and are definable relative to a first-person perspective)

I think I agree with your main point, which is that we should avoid perspective-inconsistent propositions (e.g. "I am the first clone") unless they have a clear translation to some specific perspective. I think these translations are usually pretty easy in these problems, though; they generally have a pretty obvious interpretation as a predicate on centered world models.

We agree that "I am in a blue room" is a valid proposition that an agent should assign a probability to in the cloning case, since it's entirely first-person. Here's a cloning problem analogous to sleeping beauty:

1. The experimenter flips a coin.

2. If it comes up heads, you're put in a blue room. If it comes up tails, you're cloned once (so there are 2 copies), one copy goes in a green room, and the other goes in a red room.

Before seeing what color your room is, what probability should you assign to the room being blue, green, or red? SSA says blue 1/2, green 1/4, red 1/4; SSA+SIA says each is 1/3. I don't know whether you would recommend a procedure for assigning probabilities in situations like this, and if so what probability your procedure would give here. (In any case, I am guessing that we agree that this problem is not automatically resolved by noting perspective inconsistency, as the only propositions considered are first-person)

Replies from: dadadarren
↑ comment by dadadarren · 2018-08-10T14:35:46.022Z · LW(p) · GW(p)

As for the calendar I understand your view. Treating a material object as a perspective is not something I like either, I think it's just one way to express the argument. Alternatively it can also be seen as something which its evolvement follows that the perspective of an outsider. For beauty to base her decision on the condition of that object is analogous to switching to an outsider's perspective. Like in sleeping beauty problem there are numerous arguments using monetary awards or bets. Depending on if the bookkeeping is done by an outsider or by beauty herself the conclusion would be completely different. But again my suggestion is that we do not dwell to much on this. Because whether or not the awakenings happens on specific dates, whether or not calendars are significant in the experiment setup, sleeping beauty is still the same problem. The paradox is only caused by the different number of awakenings.

My position to the problem is that the probabilities are for the blue 1/2, and others 1/4 each. As stated in my argument I shall believe the coin fall with an equal chance. So in case of heads my room would be blue the probability is 1/2. If it is tails then the room assigning experiment is exactly like the room painting problem from my previous reply. With no other information I would assume the probability of me being assigned to either room are equal. Hence 1/4 each. So far it is pretty similar to SSA's answer. However I want to point out if the question is modified that in case of tails the original would be put into the green room and the clone goes into the red room then the question becomes invalid (the red or green part, the blue part is still half.) Because in this case my room is defined in first-person as well as in third-person (original or clone). A telltale sign is that the imaginary experiment would involve soul embodiment. For such perspectively inconsistent questions there is no possible answer. But SSA and SIA would still produce an answer with no problem.

If I ask why am I a human being rather than a cow?(here "I" is a first-person definition. So I'm asking why am I experiencing the world from the perspective of a certain primate rather than a bovine.) To me it is pretty obvious logic reasoning wouldn't able to answer that. Only some sort of metaphysical conjecture or even religious creed could attempt to explain it. But with SSA there is an answer. It would be because there are more humans than cows so it's more likely that way. If we throw SIA into the mix then it can also be said because I exist it means there are a lot of humans+cows +all other kinds of minds in the universe and possibly multiverses . Of course now the reference class problem rises and the whole thing becomes messy. While in perspectively consistent reasonings I shall simply accept there is no answer and the reference class is never a problem to begin with.

Replies from: jessica.liu.taylor
↑ comment by jessicata (jessica.liu.taylor) · 2018-08-10T19:06:47.144Z · LW(p) · GW(p)

I don't know what you mean by "my room is defined in first-person as well as in third-person (original or clone)". In all problems involved, the question asked is the color of the wall in front of you, which is entirely first-person (as it is immediate in your experience). Additionally, in all problems involved, the situation you're in is affected by the actions of the experimenter, who is another person. The experimenter chooses red or green based on facts only known to them (i.e. which is the original and which is the clone), but the experimenter also chooses blue or not-blue based on facts known only to them (i.e. the coin flip). So I don't see why one question would be valid and the other wouldn't. (Note that the marble in a box problem is also of this form, where the question is entirely first-person and is affected by some other perspective, specifically my friend's)

Is there a variant of the doomsday paradox, with numbers painted in rooms, where you think asking what number is in your room is a valid question? (If so, we could set up a problem where the experimenter paints 2 numbers on each room, the first equal to the room's index (with "lower" rooms always being filled first), and the second equal to the total number of copies. Then we could ask what probability distribution a copy should assign to the second number conditional on knowing the first number. If you use SSA for this you get a doomsday prediction)

Replies from: dadadarren
↑ comment by dadadarren · 2018-08-12T06:55:31.616Z · LW(p) · GW(p)

Ok, let's ignore about how are the rooms defined. In the question I am also defined both in first-person as well as in third-person. And the difference is easier to show this way.

The difference is this: for the original problem in your previous comment the uncertainty about red or green is due to the method of assigning the colors are unknown to me. But for the problem I modified the "uncertainty" is due to that I can be either be the original or the clone. The first uncertainty is explainable within the first-person perspective. I don't know the method. Plain and simple. The "I" in "I don't know" obviously means the first-person self. While the second kind of "uncertainty" needs both perspectives to interpret. Form first-person there is no uncertainty about who I am: this is me. Distinguishing the two clones basing on their difference, like original or clone, is an outsider's logic. But if I reason from an outsider's perspective, and ask if a specific person is the original or clone then the problem is which one is this "specific person". Obviously that person is the first-person self. Effectively we need to switch perspectives to make sense of the supposed uncertainty. Hence the perspective inconsistency and the reason why I say it is invalid.

In my blog under the doomsday argument section I said asking the probability of my room number (indexed) is not a valid question. The reason is the same as above: to understand the question one needs to switch perspectives. If we keep a constant perceptive the doomsday argument fails. For example from third-person perceptive seeing my room number simply means an ordinary clone with that number exists (instead of a specific clone has that number). And that is no evidence to favour the lower population.

Replies from: jessica.liu.taylor
↑ comment by jessicata (jessica.liu.taylor) · 2018-08-12T08:18:45.827Z · LW(p) · GW(p)

Why doesn't the blue vs. not blue question require an outside perspective to interpret?

Replies from: dadadarren
↑ comment by dadadarren · 2018-08-13T00:34:20.438Z · LW(p) · GW(p)

Because to make sense of this question I do not have to think from both perspectives. In the question "I" is whoever that's most immediate to perception. So it is fully understandable within first-person perspective. Yes the room is painted by another individual but I do not have to use a theory of mind to reason from his perspective to appreciate the uncertainty.

Compare to asking the probability of me being the clone vs the original. From first-person perspective I specify myself only by immediacy to perception. Using differences to differentiate the clones (like clone or original) is third-person thinking. Therefore to understand the question I do have to think from both perspectives. This means it is a perspectively inconsistent question.

Replies from: jessica.liu.taylor
↑ comment by jessicata (jessica.liu.taylor) · 2018-08-13T02:04:36.153Z · LW(p) · GW(p)

In the red/green case the question isn't clone vs original, it's whether the wall right in front of you is red or green.

I don't see why you don't have to reason from the experimenter's perspective in the blue/not blue case, since the experimenter is the one deciding where to put people (based on the coin flip).

Basically I see two consistent positions on this set of questions:

  1. Any question about the color of the wall in front of you is valid and can be assigned a probability, since the color is definable in first-person.

  2. A question about the color of the wall in front of you is only valid if reasoning about this color doesn't require theory of mind about another person.

1 would obviously say that the red/green question is valid. 2 would say that the blue/not blue question is invalid, since whether you end up in a blue room depends on decisions the experimenter makes. (2 would also say that the color of the marble in the box can't be asked about in the marble problem)

You might have another consistent position but I am having a lot of trouble determining what it is.

Replies from: dadadarren
↑ comment by dadadarren · 2018-08-15T02:17:05.554Z · LW(p) · GW(p)

At this point I feel any further attempt to explain my position would be at the risk of repeating my previous arguments. The frustration is real because I think my idea is actually very simple. I'm having a bit of struggle to express it. From your reply I feel like we are not exactly engaging each other's argument on the same page. There must be something fundamental that the two of us are not having the same definition yet we don't realize. So I will get back to what I meant by first-person and third-person perspective as well as their differences. Maybe that way my reason of why some of the questions being invalid would be a little bit obvious to understand.

First-person perspective to me is the realization that my reasoning is based on my consciousness and perception. One of its perk is self identification based on subjective closeness to perception, which do not need any information. E.g. Twins do not need to know the objective differences between the two to tell themselves apart. For anybody else differences must be used to specify one among the pair (like older vs the younger). A question is perspectively consistent if it can be fully interpreted by one of the perspectives. For everyday probability questions, either perspective would do the job. For example, if you (I assume Jessica) and a guy called Darren are in a experiment where a fair coin is tosses. If heads then only one of the two would be waken up during the experiment whereas tails means both would be awaken. It can be asked from your first-person perspective what is the probability of me waking up in the experiment? Here "me" can be interpreted as the special person most immediate to perception. From third-person perspective, or the perspective of an outsider if you prefer, the two person are in equal positions. It can specify one by their differences (for example their names) and ask what is the probability of Jessica waking up in the experiment? Both questions are fully contained within their own perspectives. Both of which are valid.

But for a question such as "am I the clone or the original?" that's not the case. It requires first-person perspective to specify an individual by immediacy to perception while also requires third-person perspective to put the two clones in equal positions and differentiate them base on their originality. That's why it requires us to switch perspectives to attempt to understand it thus invalid.

As for the red/green question. If it is known that the original would be painted red and the clone painted green then asking what color would mine be obviously is asking if I'm the original or clone. Of course the wall in front of me is defined by proximity to perception, but the supposed uncertainty of its color is only because I don't know who I am from a third-person perspective (original/clone). So we need both perspectives to interpret the question. Compare that to blue/not blue. The wall is still define by proximity to perception and the uncertainty is due to the coin toss. I am perfectly capable of understanding what a coin toss is without having to identity me by some objective differences among a certain reference class. So that question is understandable solely from first-person perspective thus perspectively consistent.

Replies from: jessica.liu.taylor
↑ comment by jessicata (jessica.liu.taylor) · 2018-08-16T09:10:21.294Z · LW(p) · GW(p)

[EDIT: retracted]

comment by jessicata (jessica.liu.taylor) · 2018-08-05T21:08:13.437Z · LW(p) · GW(p)

One note about terminology: there is no such thing as a third-person perspective. The central case of a perspective is a point or timeline in spacetime that things can be viewed from (see: perspective drawing). A more general notion of perspective is that of a set of nodes in a causal graph from which indexical references to other graph nodes can be defined. But in any case these perspectives are necessarily first-person, since they are from somewhere.

You could try to use the ontology of physics to define hypothetical perspectives on our universe that are outside of our universe, but once precisely defined, these would also be first-person, and defining these perspectives would be complicated by issues like defining objects and interpretations of quantum mechanics.

In the sleeping beauty problem, it is sensible to talk about the experimenter's perspective, but not to talk about a non-specific third-person perspective.

(as one case of this: you talk of using names to address individuals from a third-person perspective. But, unless your names are really really long, this doesn't work in a big universe if the third-person perspective is taken to be some perspective definable from physics, since there are more individuals than short names in a big universe. It does work if by "third-person perspective" you mean some first-person perspective, e.g. that of a member of a group or some intersubjective perspective formed from those of the group members, that needs to uniquely refer to each of some not-extremely-large number of individuals who are spaciotemporally close)

EDIT: just realized that "first-person" has the implication that the perspective is from some person, i.e. a kind of mind. But not all perspectives are from a mind (e.g. camera perspectives). They are still located in a particular place in a causal graph (such as spacetime), though.

Replies from: dadadarren
↑ comment by dadadarren · 2018-08-07T01:05:34.414Z · LW(p) · GW(p)

I think you are correct that all perspectives are ultimately first-person. That's why I had to say to reason in third-person is to use a theory of mind to deduct how another person would reason. As a result I guess I'm using the term first-person and third-person in sense of everyday language rather than strict philosophical terminology. I appreciate this insight. Please forgive me in the following paragraphs as I will keep using the terms this way. Not because I don't agree with you. Just that it makes writing easier.

My main argument is that from first-person perspectives there are unique perk/limitation not applicable to any body else. Such as one do not need any information to specify oneself, and one would always find oneself exists. If I bypass these perks and limitations then my reasoning would be meaningful to others in general. For example twins do not need to know their difference to tell themselves apart. But for everybody else differentiating them require knowing their difference. For the twin it is natural to ask how would others identify me without having to specify which exact person's perspective among the "others" must he reason from to answer the question. Of course there is nothing wrong in specifying a third person either. In the sleeping beauty problem, we can take the experimenter's perspective, an observer's perspective, or even an imaginary person's perspective, just that their reasoning would be the same. I am not trying to define third-person perspective as the unique perspective of an outside-the-universe observer. As you pointed out that would make uniquely identifying any individual near (if not completely) impossible. Even worse it would make identifying the reference class impossible as well. I feel this identification would ultimately fall back to immediacy to perception. As you suggested in the names case: spatiotemporally close. Which again showing that you are correct in saying all perspective are first-person.

In the case of perspective of a non-mind such as a camera. To be completely honest I feel I do not know enough to contribute an opinion. Coming from a civil engineering background philosophy is not my forte. It is already quite difficult for me to put these not so easily describable ideas down in a second language. Can we reason from the perspective of a camera? I want to say yes? Because we can imagine it has a mind and mind is a non-physical concept so there is no logical contradiction. But again my opinion probably don't worth two cents. Just want to say that this part I don't think can change the answer to doomsday argument or the sleeping beauty problem.

comment by Radford Neal (radford-neal) · 2018-08-08T17:38:46.693Z · LW(p) · GW(p)

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).

Replies from: dadadarren
↑ comment by dadadarren · 2018-08-10T05:37:28.535Z · LW(p) · GW(p)

Professor Neal, first of all I want to thank you for your input in this discussion. I'm an U of T graduate from 2010, so getting a reply from you means a lot to me.

Of course I have read your paper on the FNC approach. As you said it treats all interference from a third-person perspective. I think we have a disagreement about why for everyday problem third-person and first-person perspectives do not cause a difference in answer. According to FNC approach it is because that we have enormous amount of details information available that there is another person with the same perception would be virtually impossible. So the answer from both perspectives should be extremely close or for all practical purposes: equal. I think the reason is because for everyday probability problems third-person identity is obvious and they are not related with questions of self-existence. The main differences between the two perspectives is that first-person identity is inherently obvious but third-person identity is not. Also from first-person perspective self-existence is a guaranteed observation while from third-person perspective no one's existence is a guarantee. So for everyday problems none of these two points matters. So the reasoning from either perspectives would be the same. And we can arbitrarily switch perspectives without worrying about a change in the answer.

I completely agree with your analysis on the island problem. The example is trying to repeat the argument of Technicolor Beauty by Titelbaum yet arriving at an obviously incorrect answer. In fact when I was writing the example I was thinking about the Boy or Girl Paradox. But it makes perfect sense that it reminds others about the Monty Hall problem. Since from both problems the key to the answer is not about what information do we have but rather about how we got the information. As you have pointed out in the island problem the process of knowing the information "the older kid is a boy" was twisted and we get a wrong answer. In the Monty Hall problem the key is not the empty door shown but that the host knows which doors are empty and is selecting among those rather than just randomly selecting a door to show us. Yet in the technicolor argument the information available is conceptualized as an static description that "I am awake on a blue day." without discussing how is that obtained. And that how is exactly where the paradox is at. If I treat me as a randomly selected individual from all actually exist individuals then seeing the color blue doesn't really matter in calculation and we get halfer's position. If I treat me as a randomly selected individual from all potential individuals then seeing blue matters and we get thirder's position. It is only due to our habit of interpreting the language that the argument concludes thirders are correct. After all, if the color does not matter why purposely say I am awake on a blue day.

In my opinion FNC is definitely superior to SIA because it does not out right uses first-person identity in an otherwise third-person argument. However it still need an assumption about how all the details of perceptions are obtained from a third-person perspective. Because the same details and information functions differently in calculation depending on the process.

In my opinion we should always keep the perspectives separate. Then there is no assumption involved. The details I see in first-person that are not relevant to the coin toss would not need to be kept in mind in our calculation.

Replies from: radford-neal
↑ comment by Radford Neal (radford-neal) · 2018-08-10T12:56:40.347Z · LW(p) · GW(p)

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?

Replies from: dadadarren
↑ comment by dadadarren · 2018-08-11T01:47:44.268Z · LW(p) · GW(p)

I think we agreed that FNC reasons from a third-person perspective, which i would say SIA attempted to do so as well. From this perspective all clones are in indifferent positions. Of course from a clone's first-person perspective the process of knowing the color was simply opening my eyes and saw a piece of blue paper. But from a third-person perspective, where no clone is inherently special, it remains a question of how come the details in discussion is from one particular clone's observation rather than from any other (potential) clone's. Here a process is missing explaining how is that clone chosen.

As you have pointed out in the island problem this process is crucial in the calculation. The fact of "older child is boy" can be the answer to the question of "Is this boy the younger or the older child" or "Is the older child a boy or a girl". Different questions imply different processes of how the fact was learnt and the calculation would be different. The island problem assumed latter question thus used the wrong process and got the absurd answer. Similarly for the technicolor beauty problem that fact that "awake on blue day" can either be the answer to "what color was assigned to this day?" or "is there an awakening on the blue day?". In the technicolor beauty argument the question was chosen to be the latter. There is no justification for this. With this question implies an imaginary process from the third-person perspective: from all days the blue day is specified, then it is checked to see if there is an awakening in it. This is the process SIA assumes in the first-place. Of course its conclusion would confirm thirder's answer. So the technicolor beauty is just showing SIA would lead to a thirder conclusion, nothing more. In another word, only thirders should conclude the probability of heads to be 1/3 after considering the color of the paper. The argument attempts to show even if someone initially assigns equal probability to heads and tails he should update his answer to 1/3 after seeing the paper. It is incorrect. For example a supporter of SSA would say beauty after waking up must be in one of the three positions: 1H,1T, 2T (here the number of 1 or 2 means first or second day and H or T means heads or tails.), the respective probability according to SSA is 1/2, 1/4, 1/4. Regardless of which situation she's in the likelihood of seeing blue would always be 1/2. So he would still conclude the probability of heads as 1/2. That is because according to SSA the process of learning about the blue paper (from third-person perspective) is different. Here an awakening is first chosen among all awakening(s) and the color of that day just turns out to be blue by chance. Applying technicolor beauty's argument in this case and say he should update to 1/3 would be making the exact mistake the island problem did. In effect after considering the paper color thirders would still be thirders and halfers should remain halfers. Meaning the color is inconsequential to the problem.

I agree using all information available, though not necessary in most cases, would give the correct answer. But here the process of which clone's detailed observation is chosen to be used in third-person argument, which is the key information to calculation, is assumed. Then it is no longer safe to say FNC must be correct. In my opinion the supposed missing process is trying to link first-person and third-person perspectives. The link would cause perspective inconsistency thus there should be no such process to begin with. The perspectives should just be kept separate.

As for the so called "Ad hoc" information it's my mistake to just use made up terms and not defining them. When we deal with everyday problems there are always some detailed information with no effect one the answer that we automatically ignore them in the calculation. These are details that are not related to the subject matter at hand, eg being the older kid has no effect on the sex of the child; and played no part in the process of how we get to know relevant evidences, e.g. being the older kid does not change the chance of him coming to the door. These are what I refer to as Ad-hoc informations because they cannot be pre-specified in an observation. As in the island problem the kid at the door just happens to be the older one. If I was predetermined to meet the older child then I have to use this info and specify him as such in the calculation and the answer should then be rightly half. Other example of Ad-hoc information could include how does the boy look, what is he wearing, which day of the week was he born in or any other detailed information you can get about him. I think it is best practice if we just ignore these. As using these info to specify the boy in front of you would lead to mistakes in calculation. But as you have shown in the first reply they can be used if we pay attention to the process of how are these information learnt (or what questions does these details answer). So I am mistaken to say these info cannot be used. Just that correctly using these information would not make any changes to the answer.

Replies from: dadadarren
↑ comment by dadadarren · 2018-08-13T01:11:06.986Z · LW(p) · GW(p)

I think I can put more structure into my argument comparing the island problem to technicolor beauty.

The Island Problem

While the statement "the older child is a boy" is factually true it can be learnt by two different processes.

Process 1:

First a boy is specified among all boy(s). One can ask: "is this boy the younger or the older child?". Then it found out that he is actually the older.

Process 2:

First the older child is specified among all children. One can ask: "is the older child a boy or a girl?". Then it found out that the older child is actually a boy.

As you have pointed out in the first reply the correct process is Process 1. However in the island problem the calculation was done according to Process 2. That is why its answer is wrong.

The Technicolor Problem

While the statement "I'm awake on a blue day" is factually true it can also be learnt by two different processes.

Process 1:

First an awakening is specified among all awakening(s). One can ask: "is this awakening a blue or a red awakening?" As beauty opens her eyes it is found out that it's the blue one.

Process 2:

First the blue day is specified among all days. One can ask "is there an awakening on the blue day?". Then it is found out that there is indeed an awakening on the blue day.

Technicolor beauty used Process 2 in its calculation without any justification. To me Process 1 is describing what actually took place. Before opening my eyes, I can ask "is this awakening red or blue?" and expect to find the answer after opening my eyes. I cannot ask "is there an awakening on the blue day?" and expect to find an answer. What if the paper turns out to be red? Shall I retrospectively change the question to ask about the red day instead?

To me the justification would be treating today as a randomly chosen day among the two days. Then Process 2 would be the correct description. However that is exactly what SIA assumes in the first place. SIA would lead to thirder's answer regardless if there are papers involved. People thinking the coin fell with equal chance would disagree and say Process 1 is the correct one to use. Using which their probability, even after considering the papers, would still remain at half. So the added detail of different colors would be inconsequential to the problem after all.

Replies from: radford-neal
↑ comment by Radford Neal (radford-neal) · 2018-08-13T20:57:23.075Z · LW(p) · GW(p)

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.

Replies from: dadadarren, jessica.liu.taylor
↑ comment by dadadarren · 2018-08-15T08:32:31.867Z · LW(p) · GW(p)

I see what you mean. I agree that we know more than just "the older child of the family is a boy". The "more" would be the process of how I come to know it. To me what's special about the island problem is that when trying to express what I know into a simply statement such as "the older child is a boy" any information about the process is lost. Therefore it left us with an ambiguity about the process thats up to interpretation. This is exactly what happens in the Boy or Girl paradox as well. If there is any lesson then it should be conditioning on a statement such as "someone with all those detailed perception and memory exists" is a rather delicate matter. Is this someone specified first and then all the details about her explored? Or is all these details spelled out first and someone with these details was found to be exist? SSA and SIA would give different answers from a third-person perspective. But from first-person perspective the process is clear. It is the former. That someone is specified based on immediacy to perception, i.e. that someone is this one. And then all the details about me are found out though my experience. Therefore the perspective consistency argument would not change its answer basing on any details observed after waking up.

As for the disagreement, more preciously the "agree to disagree", between friends while in communication. I'm aware it is a rather peculiar case. SIA and FNC would not result in that which can certainly be used as a argument favouring them. But in my opinion it can be quite simply explained by perspective differences. Of course basing on my experience with paradoxes relating to anthropic reasoning, nothing is simple. So I understand if others find it hard to accept.

Replies from: radford-neal
↑ comment by Radford Neal (radford-neal) · 2018-08-15T16:29:44.977Z · LW(p) · GW(p)

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.)

↑ comment by jessicata (jessica.liu.taylor) · 2018-08-14T08:31:58.169Z · LW(p) · GW(p)

SSA conditions on more "information" than that an observer with your observations exists; specifically, it conditions on the fact that a randomly selected observer has your observations, which automatically implies that an observer with your observations exists. (I put "information" in quotes because this is only information if you accept something like SSA)