I The Sleeping Beauty Problem: Any halfers here?

[Dale]

She should state her "credence" as 2/3 and hope the experimenter accepts that number as her credence for the event "Today is Monday".

As you have described it, that is in fact her credence since it gives the price she would buy or sell the defined bet.

 

[Dale]

"Word for word" isn't sufficient to show two bets are the same. The expected payoff needs to be the same.

Every bet has the same expected payoff. Procedurally they can all be done the same too. They are the same linguistically, financially, and procedurally.

 

[Stephen Tashi]

As you have described it, that is in fact her credence since it gives the price she would buy or sell the defined bet.

We're going round and round. The experimenter offers SB a bet stated as "If the coin landed heads you win $1". On Monday, this is fair bet. On Tuesday, it is a sure loser. SB knows that if she says she is wiling to pay X for the bet, she will do so on both bets whenever the second bet is offered. If you want to define a bet to measure SB's credence for the event "the coin landed heads" then please define a bet where she does not incur the possibility of having to buy a second bet with a different expected payoff.

 

[Dale]

The experimenter offers SB a bet stated as "If the coin landed heads you win $1". On Monday, this is fair bet. On Tuesday, it is a sure loser. SB knows that if she says she is wiling to pay X for the bet, she will do so on both bets whenever the second bet is offered.

Yes. So she had better factor all that into the price she is willing to pay for the bet.

If you want to define a bet to measure SB's credence for the event "the coin landed heads" then please define a bet where she does not incur the possibility of having to buy a second bet with a different payoff

The bet to measure her credence is already defined. No exceptions are listed in the definition.

Stop trying to change the definition, just use it as stated.

 

[Stephen Tashi]

Every bet has the same expected payoff.

No.

If SB were awakened at random days in random experiments, the bet "You get $1 if the coin lands heads" could be assigned a single expected payoff because the 3 situations where the bet happens are random variables chosen by "random sampling with replacement"

However, in computing the betting strategy, the systematic plan of the experiment is considered. After the coin toss, the situations that arise are determined, not random.

The bet to measure her credence is already defined. No exceptions are listed in the definition.

Here's a bet: "You get 1$ if the coin lands heads - and we toss another coin and you lose $10 if it lands heads".

Do you really think the fair price for that bet is the same as the fair price for "You get $1 if the coin lands heads"?

 

[andrewkirk]

In post 384 I have several links I found useful. The second one has a brief definition that we have seemed to settle on.

The Sleeping Beauty Problem: Any halfers here?

thanks for that. From the Stanford article we get the following definition, which I'm reproducing here so everybody can see it without having to follow links:

Your degree of belief [credence] in E is p iff p units of utility is the price at which you would buy or sell a bet that pays 1 unit of utility if E, 0 if not E.

With that in place, we now need to know exactly what SB is betting on. In particular, we need to know whether she knows, before the experiment begins, that she will be offered a bet every day or, if not, on what basis it will be determined whether she will be offered a bet. If she does not know that, and remember it each time she wakes up, none of the analysis I've read so far about betting applies.

Let's assume that she does know, and always remembers, that each day she will be offered the opportunity to pay $x$ dollars for a chance to name Heads or Tails and then immediately win $1 if what she named was correct, and also that $x$ will be the same every day! Then it's a simple exercise in expected values of net payoffs, as per my previous post.

The expected payoff, over the course of the entire experiment, of a strategy that always guesses Heads (recalling that Heads is associated with only being woken on Monday, on the wiki presentation) is (giving the probability for the case where the coin was actually Heads in the first term):
$$0.5\times (1-x)+ 0.5\times 2 \times(0-x)=0.5-1.5x$$

So, for the bet to be worth taking, it must have an expected payoff greater than 0.

For Heads, this means that $0.5-1.5x>0$ so that $x<1/3$

If SB is offered the Heads bet with $x=0.33$, and remembers the full details of the betting program, including that she'll be offered the same bet, at the same price, every day, she will take the bet.

So on that definition, and with those very specific rules about betting, (which were not stated in the original problem statement), we might say that the credence in the proposition that the coin came up Heads is one third.

A different conclusion is reached if SB has not been told that she will be offered a bet every day. Then she cannot do the above calculation. Her expected payoff will be the expected value
$$0.5\times (1-x)\times P_1+ 0.5\times (0-x)\times P_2+ 0.5\times (0-x)\times P_2P_3+ 0.5\times (0-x)\times (1-P_2)P_4
$$

where $P_1$ is the probability of being offered a bet on Monday, given that the coin landed Heads,
$P_2$ is the probability of being offered a bet on Monday, given that the coin landed Tails, and
$P_3$ is the probability of being offered a bet on Tuesday given that a bet was offered on Monday, and that the coin landed Tails.
$P_4$ is the probability of being offered a bet on Tuesday given that a bet was NOT offered on Monday, and that the coin landed Tails.

This is positive if $x<\frac{P_1}{P_1+P_2+P_2P_3+(1-P_2)P_4}$. SB would make an assumption for these three probabilities in order to decide whether to bet. One possible assumption is that the probability of being offered a bet on any day is the same as any other day and independent of whether bets have been offered on other days. If so, the credence is 1/3 again.

However, an equally plausible is the assumption that $P_1=P_2=P_4$ but that only one bet will be offered, so that a bet will not be offered on both Monday and Tuesday. In that case $P_3=0$ and the credence is
$$\frac{P_1}{3P_1-P_1{}^2}=\frac1{3-P_1}$$
which will be somewhere between a third and a half.

So it depends on SB's knowledge of what betting opportunities will be made available to her.

 

[Stephen Tashi]

we might say that the credence in the proposition that the coin came up Heads is one third.

That's a subject of debate in recent posts. SB can calculate her betting strategy using P= 1/2 and never worry about calculating P(heads | awakened). In fact P(heads|awakened) has no unique value and is irrelevant to planning a betting strategy. So SB can determine what she ought pay for the bet "The coin lands heads" without computing a posterior probability that the coin lands heads. Her fair price for the bet is dishonest as a report of her credence because, according to the definition of "credence", she should imagine a bet where "You get $1 if the coin lands heads" is bet with no strings attached. However, in computing her betting strategy, as you illustrated, she considers that it might obligate her pay price X for another bet on Tuesday that is worded the same, but is a sure loser.

 

[Ken G]

Asume she pays X for the bet "You get $1 if today is Monday" and must do this every time the bet is offered. The bet is offered each time she is awakened. Taking P(Heads) = 1/2, her expected net expenditure is (1/2)X + (1/2)2X = (3/2)X. Her expected gain is (1/2)1 + (1/2)(1) = 1. She should state her "credence" as 2/3 and hope the experimenter accepts that number as her credence for the event "Today is Monday".

Of course if she uses P(Heads) = 1/3, she computes a different fair price for the bet - a wrong one, I think.

But you are mixing up the probability of heads in one way of framing the calculation, with her credence, which can be shown to be quite different. Most obvious is the 99 day version. When you say P(Heads)=1/2, you are talking about the probability that heads was flipped originally, which is of course indeed 1/2 (but that's not her credence, as we shall see). So in a single 99-day experiment, her expected expenditure is (1/2)X+ (1/2)99X = 50X. Her expected gain is 1, as you say. Ergo, she should state her credence that it is Monday as 1/50, as X=1/50 is the fair bet that it's Monday-- that is what you would get and that's correct. The problem is, the use of P=1/2 in that calculation is simply not her credence that heads was flipped, that's the probability of a heads flip taking the experiment from a kind of disinterested external perspective, analyzing the situation from the start (perhaps, from the perspective of the experimenters trying to figure out if they will bilk money out of SB). What you are not seeing yet is that if she has a credence of 1/50 that it is Monday (which we have agreed is true), and the only possible way it could be any other day is if the coin flip was tails, this clearly implies she strongly suspects the coin flip was tails, every time she is awakened. That simple truth is quite independent from the use of P(heads) = 1/2 in the above calculation.

To see the calculation from SB's perspective, it looks like this. Her credence that it is Monday is 1/50, and if it is Monday, her credence that the coin was heads is 1/2. That contributes 1/100 to her credence that the coin is heads. Her credence that it is any other day is 49/50, and if it is any other day, her credence that the coin was heads is 0. Hence her total credence that the coin was heads is 1/100, not 1/2. Furthermore, she can correctly determine her expected payoff on her bet that it is Monday by taking X = 1/50. So in the long run, for the 99 day version, she can break even if she always pays 1/50 for the bet "today is Monday" at $1 payoff, and she can also break even if she always pays 1/100 for the bet "the coin was heads" at $1 payoff. If she pays 1/2 for that latter bet, she is a chump, because her expected payoff is (1/2)*1, but her expected cost is 1/2*1 + 1/2*99 = 50. She's losing $49 every time she accepts that strategy, this is an error. So notice that even though the 1/2 appears in the calculation I just did, that's the probability of heads taken from the start of the experiment, it is not her credence on the bet "it was a heads." That's clear because if she thinks it's her credence, she gets bilked, and that proves it is not the correct credence. You should see the contradiction in thinking her credence that it is Monday is 1/50, but her credence that the coin was heads is 1/2.

 

[Ken G]

So it depends on SB's knowledge of what betting opportunities will be made available to her.

But that's obvious, it would be like playing poker and not being sure the dealer isn't dealing from the bottom of the deck. Of course that will mess up your chances, but that's just clearly not the way poker is played, it would be a cheat. SB only needs to know she is not being cheated, just as any poker player must assume, it goes without saying. This is also why the correct answer to the question is not "it depends on the precise formulation," because if you are playing bridge and you want to know the chances of dropping the queen if the defense holds 4 cards, you can look up that credence in a bridge book, and I guarantee you that you will never see "the answer depends on whether or not the dealer was cheating." SB must be told what the betting strategy is, and it must be the truth, and if that betting strategy is that she will be offered the bet every time she wakes, her credence is 1/3-- it's not an assumption, anything else would be a cheat.

 

[andrewkirk]

SB only needs to know she is not being cheated, just as any poker player must assume.

only needs to know that, in order to reach what conclusion?

If the conclusion is to be that she is willing to pay 33c for a bet that pays $1 if the coin landed Heads and zero otherwise, she needs more information than that - specifically, what are the probabilities of me being offered the same bet on every other day.

I'm afraid I know nothing about poker, so I can't really assess the extent to which the rest of your post relates to the SB situation.

 

[Stephen Tashi]

But you are mixing up the probability of heads in one way of framing the calculation, with her credence, which can be shown to be quite different.
.

Before we get into the details, let me clarify my viewpoint. My viewpoint is that the posterior probability of heads P(heads | SB is awakened) cannot be calculated from the information in the problem. Asking for it's value is an ill-posed problem and is irrelevant for planning a betting strategy. As you pointed out, planning a betting strategy should use P(Heads = 1/2)., not P(Heads|wake). The value P(heads)= 1/2 is used regardless of whether one is a "halfer" or "thirder" or agnostic about P(heads|SB awakened).

In my view, when SB answers "What is your credence that the coin landed heads?", she gives a dishonest answer if she reports the number based on her betting strategy. However, many posters in this thread are willing to accept her report as her credence that the coin landed heads, so they don't object to it. Her report is dishonest because she is not considering a simple bet "You get $1 if the coin lands heads". Instead, when planning her strategy, she considering that paying a price X for the bet obligates her to pay that price every time the bet is offered. Two different versions of the bet might offered. On Monday, it is an even bet. On Tuesday (if offered) the bet is a sure loser.

Since calculating P(Heads| awake) is an ill-posed problem, SB cannot offer a "credence" for the event (Heads | awake) unless she makes some assumptions. She does not need to calculate P(Heads| awake) unless the experimenter is stickler and can propose a bet on "You get $1 if the coin lands heads" that is a "pure" bet - i.e. a bet with no conditions that she might have to buy another bet with different expected payoff. (I myself haven't been able to formulate a "pure" bet. that could be offered to SB during the experiment.)

From my reading, the general opinion of those who have studied the SB problem is that one cannot distinguish between the "halfer" and "thirder" positions by bets that can be offered during the experiment, provided we assume SB is rational and plans a betting strategy that is independent of her opinion of P(Heads|awake).

Both "thirders" and "halfers" are incorrect to assert that their answer for P(Heads|awake) is the unique correct answer. Both the "thirder" and the "halfers" are correct that that one can create a probability model that is consistent with their answer for P(Heads|awake) and does not contradict the information given in th SB problem. The computation of a betting strategy is independent of such a probability model.

 

[Stephen Tashi]

So it depends on SB's knowledge of what betting opportunities will be made available to her.

Most posters in this thread ( including myself) assume that when she is awakened SB knows the how the experiment is conducted and that she will be asked about her credence for "the coin landed heads" every time she is awakened - but you're right, the Wikipedia statement of the problem doesn't make this crystal clear. It does say that the amnesia drug makes her forget "her previous awakening". It doesn't say she forgets other facts.

 

[Ken G]

Before we get into the details, let me clarify my viewpoint. My viewpoint is that the posterior probability of heads P(heads | SB is awakened) cannot be calculated from the information in the problem. Asking for it's value is an ill-posed problem and is irrelevant for planning a betting strategy. As you pointed out, planning a betting strategy should use P(Heads = 1/2)., not P(Heads|wake). The vale P(heads)= 1/2 is used regardless of whether one is a "halfer" or "thirder" or agnostic about P(heads|SB awakened).

Yes, it seems that the whole concept of "probability" is creating problems, but the correct betting odds should be clear enough-- 1/3 heads is the break-even betting strategy, and that's all "credence" means.

In my view, when SB answers "What is your credence that the coin landed heads?", she gives a dishonest answer if she reports the number based on her betting strategy.

Let us simply define credence by her fair betting strategy, that's really all the concept is supposed to entail.

On Monday, it is an even bet. On Tuesday (if offered) the bet is a sure loser.

Yet that is always true. If you are playing bridge, and you analyze a finesse as having a 50-50 chance, of course in one deal, it's a sure bet, and another, it's a sure loser. But that's just not what credence means, credence is not some kind of "actual probability", it is simply the correct betting strategy given the information you have. It makes no difference if she takes the bet every time or not, the correct betting strategy is always 1/3 heads because she never gets any additional information in one trial versus another. You don't have to finesse every time you get a certain hand in bridge-- yet the odds are still 50-50 if that's all you know. I suspect the problem in this discussion is the halfers are seeking some kind of "actual probability" of heads, and indeed we both just did a calculation using P(heads)=1/2, but that isn't what credence is-- credence is simply the fair bet, any time the bet is made given no information beyond what is supplied in the experiment.

From my reading, the general opinion of those who have studied the SB problem is that one cannot distinguish between the "halfer" and "thirder" positions by bets that can be offered during the experiment, provided we assume SB is rational and plans a betting strategy that is independent of her opinion of P(Heads|awake).

This is what is false. There is clearly a single break-even betting strategy in the experiment as described, and it is clearly 1/3 heads. The 99 day version makes this very clear-- you already agreed the fair bet in that experiment is X=1/50 for "today is Monday", so how can the fair bet be 1/2 heads when heads can only pay off on Monday?

 

[Ken G]

only needs to know that, in order to reach what conclusion?

In order to reach the conclusions that the fair bet is 1/3 "the coin was heads," and the fair bet is 2/3 "today is Monday."

If the conclusion is to be that she is willing to pay 33c for a bet that pays $1 if the coin landed Heads and zero otherwise, she needs more information than that - specifically, what are the probabilities of me being offered the same bet on every other day.

But the problem does clearly specify this. Using the Wiki version:
"Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: On Sunday she will be put to sleep. Once or twice, during the experiment, Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening. A fair coin will be tossed to determine which experimental procedure to undertake: if the coin comes up heads, Beauty will be awakened and interviewed on Monday only. If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday. In either case, she will be awakened on Wednesday without interview and the experiment ends."

So there it is, SB is interviewed every time, so her credence is about a bet she is given every time. It's clear as a bell, no additional assumptions needed. Of course the experimenter cannot couple the offering of the bet to the outcome of the coin, that would clearly be cheating and would do violence with the entire concept of credence. If credence were dependent on the uncertain and dishonest wiles of the experimenter, then you are saying the odds of winning any game of chance (the details of poker are of course irrelevant) cannot be known unless you know you are not being cheated. That goes without saying, it is simply what "odds" mean. If you assess the chances of a sports team winning, you never say "assuming none of the players are throwing the game intentionally," because that also goes without saying in the concept of odds.

 

[Stephen Tashi]

you already agreed the fair bet in that experiment is X=1/50 for "today is Monday", so how can the fair bet be 1/2 heads when heads can only pay off on Monday?

I'm not sure which experiment you're talking about. Which post was it?

 

[andrewkirk]

The experiment already stipulates that SB is given the same bet every time, that's what credence means

Not according to the SEP definition. There is no mention of 'every time' in that definition, because it only envisages one bet. This situation has the potential for two bets, where the number of times the bet is taken is correlated with the result of the bet. That is a completely different situation from the one envisaged in the SEP definition, which implicitly assumes that there is only one bet. Hence there is room for multiple different interpretations as to how the single-bet SEP definition might be generalised to apply to this case.

For convenience, here is the SEP definition again:

Your degree of belief [credence] in E is p iff p units of utility is the price at which you would buy or sell a bet that pays 1 unit of utility if E, 0 if not E.

 

[stevendaryl]

The error in the Thirder position is to assume that the three probabilities are the same. There is no mathematical rule to justify that.

I would say that's a conclusion, not an assumption. If you repeat the experiment many, many times, for many, many weeks, then, letting N be the number of weeks, then on the average:

• There will be N/2 times when Sleeping Beauty is awake and it's Monday and it's heads.
• There will be N/2 times when SB is awake and it's Monday and it's tails.
• There will be N/2 times when SB is awake and it's Tuesday and it's tails.
• There will be N/2 times when SB is asleep and it's Tuesday and it's heads.

So the three possibilities: (Monday, Heads), (Monday, Tails), (Tuesday, Tails) occur equally frequently. So if Sleeping Beauty's credence for each possibility is the same as the relative frequency of that possibility in repeated experiments, then she has to give equal likelihood to those three possibilities.

That's the relative frequency argument: the relative frequency of heads among the events in which she is awake is 1/3.

 

[stevendaryl]

To me, the halfer position has a very implausible consequence. There are three possibilities given that Sleeping Beauty is awake:

1. (Heads, Monday)
2. (Tails, Monday)
3. (Tails, Tuesday)

The subjective probabilities of the three, given that she is awake, and given that today is either Monday or Tuesday, must equal 1. So

P(H & Monday | Awake) + P(T & Monday | Awake) + P(T & Tuesday | Awake) = 1

So if P(H & Monday | Awake) = 1/2, and P(T & Tuesday | Awake) > 0, then it follows that

P(H & Monday | Awake) > P(T & Monday | Awake)

This in turn implies:

P(H & Monday) > P(T & Monday)

which implies:

P(H | Monday) > P(T | Monday)

This is what's bizarre to me about the halfer position. They want to say that P(H | Awake) = P(H), because being awake doesn't tell you anything new about whether it's heads or not. But the consequence of that position is that P(H | Monday) > P(T | Monday). If being awake tells you nothing about whether it's heads or tails, why would you say that it being Monday tells you something? How can the fact that it's Monday make heads more likely than tails?

What's especially bizarre about this is that, as far as the thought experiment goes, it doesn't make any difference whether you flip the coin on Monday morning or on Tuesday morning, since it's not necessary to consult the result until Tuesday morning. So in the case where the coin flip is on Tuesday morning, the halfer position implies that:

• If Sleeping Beauty is told that today is Monday, and that tomorrow morning, a coin will be flipped, she will say that it is more likely that the result will be heads than tails.

That's completely bizarre.

 

[Dale]

we need to know whether she knows, before the experiment begins, that she will be offered a bet every day

The bet is implied by the definition of credence. So it is every time she is asked about her credence, which is every interview.

So on that definition, and with those very specific rules about betting, (which were not stated in the original problem statement)

They were stated in the original problem. Beauty is asked about her credence in each interview so the implied bet is necessarily offered each interview also.

 

[stevendaryl]

Not according to the SEP definition. There is no mention of 'every time' in that definition, because it only envisages one bet. This situation has the potential for two bets, where the number of times the bet is taken is correlated with the result of the bet. That is a completely different situation from the one envisaged in the SEP definition, which implicitly assumes that there is only one bet. Hence there is room for multiple different interpretations as to how the single-bet SEP definition might be generalised to apply to this case.

I'm not so sure that it matters. It is true that the number of opportunities to bet depends on the coin result. However, each individual opportunity to bet seems to fit the definition.

 

[Buzz Bloom]

The probability of being woken on Monday in the Tails situation is 1/2 the probability of Tails, which is 0.5 times 0.5 = 0.25. The same goes for Tuesday-Tails.

Hi:andrew:

You seem to be making the same mistake that I made before I recignized my mistake as I explain in my post # 498.
There are two awakenings for tails, not one.

Regards,
Buzz

 

[Dale]

posterior probability of heads P(heads | SB is awakened) cannot be calculated from the information in the problem.

I have shown that this is false back in post 255. At this point, I would ask you to stop repeating this false claim.

she gives a dishonest answer if she reports the number based on her betting strategy. ...The computation of a betting strategy is independent of such a probability model.

And this is a mistake on your part. The honest price that she would buy or sell the bet is the value of her credence. If she answered with a value that she would not bet on then she would be dishonestly representing her credence

 

[Ken G]

I'm not sure which experiment you're talking about. Which post was it?

Take the correct calculation of X you did in post 549 and apply it to the version of the puzzle with 99 days. You should get X=1/50, meaning, SB would take 49 to 1 odds against it being Monday.

 

[Ken G]

Not according to the SEP definition. There is no mention of 'every time' in that definition, because it only envisages one bet.

The puzzle stipulates that she is interviewed every time. That means she is given the same interview every time, because to assume otherwise is to assume dishonest collusion on the part of the experimenters. If they give her a different interview to try to fool her into making a bad bet, they are cheating. It goes without saying in any puzzle that the experimenters are not intentionally cheating.

This situation has the potential for two bets, where the number of times the bet is taken is correlated with the result of the bet.

There is a bet in every interview, that's what establishing the credence, in the interview, means, according to the definition you gave and I am also using. I see that Dale is making that same point.

 

[.Scott]

The sleeping beauty problem is a well known problem in probability theory, see e.g.
https://en.wikipedia.org/wiki/Sleeping_Beauty_problem
http://allendowney.blogspot.hr/2015/06/the-sleeping-beauty-problem.html
https://www.quantamagazine.org/solution-sleeping-beautys-dilemma-20160129
or just google.

I believe we need to examine the judgement and perhaps the intentions of these experimenters more closely.
Did these experimenters dupe Sleeping Beauty into taking a dangerous cocktail of drugs in the name of science?

It is the amnesia drug in particular that is of most concern. These drugs, such as Valium and high doses of alcohol are potentially addictive and are associated with suicide. Assuming these experiments are of sufficient value to science (and I am sure they are), they should be designed for minimal use of these drugs.

However, it is clear from the methodology that no such consideration was given.
Why was amnesia induced after Sleeping Beauty responded to the questions on Tuesday?
When the coin came up heads, why was amnesia induced at all?
In fact, once the last interview was conducted on either Monday or Tuesday, why were any additional drugs used at all?

I will also note that this experiment was designed in 2000, decades after the over-prescription of Valium had been widely publicized.

 

[Ken G]

All humor aside, that is actually a good point-- there's no need for amnesia drugs for a heads flip. The answer to the puzzle, 1/3 credence for heads, is the same as long as SB knows she would have forgotten Monday's events had tails been flipped, nothing else is needed. The puzzle specifically says she only forgets the awakening, so the drugs are only needed after the Monday interview and only if tails came up (Returning to humor vein, at least the way it is set up allows the experiment to be done double-blind, though of course the interviewers will also need to be given the amnesia drug, further complicating the ethical dilemma.)

 

[Stephen Tashi]

I'm not so sure that it matters. It is true that the number of opportunities to bet depends on the coin result. However, each individual opportunity to bet seems to fit the definition.

How can an answer that Sleeping Beauty obtains without considering the probability of ( heads| awake) indicate her credence of that event?

The bet (heads'awak) on Monday is even odds. If offered, the bet on (heads | awake) on Tuesday is sure loser. Just because Sleeping Beauty doesn't know which bet she is making doesn't mean they are "the same to her". Her calculations for what to bet don't involve computing the probability of (heads | awake). She simply uses the fact that p(heads|) = 1/2 and accounts for the probability that she may be for to make two different bets. If she was treating the bets as being the same, her calculations would show they had the same pay-off.

 

[Stephen Tashi]

Take the correct calculation of X you did in post 549 and apply it to the version of the puzzle with 99 days. You should get X=1/50, meaning, SB would take 49 to 1 odds against it being Monday.

SB is assume to be rational. A rational "halfer" or a "thirder" doesn't use their estimate of p(heads | awake) to decide on their bets. What SB bets should be iindependent of p(heads|awake).

 

[Stephen Tashi]

I have shown that this is false back in post 255. At this point, I would ask you to stop repeating this false claim.

Request denied. The claim isn't false. Your two-line derrivation of the "thirder" claim is not a valid proof. If the "thirder" answer was unique , the "halfer" answer would not be another solution.

And this is a mistake on your part. The honest price that she would buy or sell the bet is the value of her credence. If she answered with a value that she would not bet on then she would be dishonestly representing her credence

SB computes her answer without computing P(heads | awake). In fact, her answer is computed using p(heads) = 1/2. So how does her answer indicate her credence for the event p(heads | awake)?

A rational "halfer" gives the same answer a "thirder" does, so how does credence in P(heads | awake) have anything to do with the bet?

 

[stevendaryl]

How can an answer that Sleeping Beauty obtains without considering the probability of ( heads| awake) indicate her credence of that event?

I don't understand the question. You wake Sleeping Beauty up. In case she doesn't remember, you remind her of the rules. Then you ask her if she wants to bet on whether it's heads or not. Of course, she will consider the probability of heads given that she's awake in deciding which way to bet.

 

[Stephen Tashi]

I don't understand the question. You wake Sleeping Beauty up. In case she doesn't remember, you remind her of the rules. Then you ask her if she wants to bet on whether it's heads or not. Of course, she will consider the probability of heads given that she's awake in deciding which way to bet.

No.

A rational Sleeping Beauty will know that if she always answer X then the expected net price she must pay for doing so is: (1/2)X + (1/2)(2X) = (3/2)X. Her expected gain from giving that answer is (1/2)(1) + (1/2)(0). So she solves (3/2)X = (1/2) to obtain X = 1/3. This calculation is done using P(Heads) = 1/2 and does not involve computing p(Heads | awake).

 

[stevendaryl]

No.

A rational Sleeping Beauty will know that if she always answer X then the expected net price she must pay for doing so is: (1/2)X + (1/2)(2X) = (3/2)X. Her expected gain from giving that answer is (1/2)(1) + (1/2)(0). So she solves (3/2)X = (1/2) to obtain X = 1/3. This calculation is done using P(Heads) = 1/2 and does not involve computing p(Heads | awake).

I don't understand the distinction you're making. I would say that that calculation IS the calculation of P(H | Awake). Sleeping Beauty knows that if the experiment were repeated many times, then approximately 1/3 of the times in which she is awake will be when the coin toss was heads, and 2/3 of the times in which she is awake will be when the coin toss was tails. If you want credence to line up with relative frequency, then the credence of heads, given that she is awake, should be 1/3.

 

[Ken G]

SB is assume to be rational. A rational "halfer" or a "thirder" doesn't use their estimate of p(heads | awake) to decide on their bets. What SB bets should be iindependent of p(heads|awake).

Please take your post 549 and use it to calculate, via the very method you used there, the X for it being Monday in the 99 day version. You should get X=1/50, just do the same thing you did in post 549. Is that indeed what you get?

The bet (heads'awak) on Monday is even odds. If offered, the bet on (heads | awake) on Tuesday is sure loser. Just because Sleeping Beauty doesn't know which bet she is making doesn't mean they are "the same to her". Her calculations for what to bet don't involve computing the probability of (heads | awake). She simply uses the fact that p(heads|) = 1/2 and accounts for the probability that she may be for to make two different bets. If she was treating the bets as being the same, her calculations would show they had the same pay-off.

There are two different ways SB can arrive at her credence that the day is Monday, one involving P(heads)=1/2 from the start, and another involving P(heads/awake)=1/3. (That second calculation asserts that her credence that it is Monday equals P(heads|awake) + P(tails|awake)*1/2) That both give the same answer shows that the correct credence that it is Monday is 2/3, which you got using the first calculation involving P(heads) rather than the second calculation involving P(heads|awake). But note that P(heads)=1/2 is not her credence that the coin is heads, P(heads/awake)=1/3 is. This is the mistake you are making.

 

[Stephen Tashi]

I don't understand the distinction you're making. I would say that that calculation IS the calculation of P(H | Awake).

SB would make the same calculation if she's a "halfer"

Consider this experiment:: A coin is flipped. A red hat is put on your head.. If the coin lands heads, you will be asked "What is a fair price for the the bet that you get $1 if the coin landed heads?" You give some answer X and pay that amount to purchase the bet. If the coin landed tails, you are required to purchase the bet again at the same price.

Is the answer X that you choose to give equal to your creedence for the event "The probability that (the coin landed heads | given I'm wearing a red hat)?

 

[Ken G]

The whole discussion comes down to this. Given the rules of the scenario, where SB is interviewed every time she is wakened (not just some arbitrary set chosen by cheating experimenters), we may conclude that every time she is interviewed, it is true that P(heads)= 1/2 and P(heads|awake)=1/3. That's the halfers and thirders right there. The thirders are correctly responding to the question asked, which is, what is her credence given that she has been wakened. The halfers are answering the wrong question, they are simply asking what is the probability that the coin came up heads in the first place, which is not a probability that can be altered.

I think it becomes clear what the halfers are thinking if we consider a 99 day version, but we make the following change. If heads are flipped, SB is interviewed only on Monday. If tails, SB is interviewed on a single randomly chosen day, sampled equally from the 98 days after Monday. In that case, her credence that it is heads is clearly 1/2, that's the halfer thinking. The halfers are claiming her credence is not affected if we change the experiment to interview her all 99 days if it's tails. That's clearly wrong, but to make it perfectly clear, we can simply look at her credence that it is Monday in the two situations. The halfer thinks her credence it is Monday is 1/2 in both versions of the 99-day experiment, because if the coin flips a heads, she gets interviewed on Monday. But that's not half the times she is awakened.

 

[Stephen Tashi]

To me, the halfer position has a very implausible consequence. There are three possibilities given that Sleeping Beauty is awake:

1. (Heads, Monday)
2. (Tails, Monday)
3. (Tails, Tuesday)

The subjective probabilities of the three, given that she is awake, and given that today is either Monday or Tuesday, must equal 1. So

But the consequence of that position is that P(H | Monday) > P(T | Monday). If being awake tells you nothing about whether it's heads or tails, why would you say that it being Monday tells you something? How can the fact that it's Monday make heads more likely than tails?

"Halfer" answers will seem bizare if you don't think in terms of proability model that is consistent with "halfer" answer.

Compute the probabilities using the "halfer" probability model that I've mentioned before. One probability model consistent with "halfer" viewpoint describes how to select from the 3 events you mentioned in the following manner:

1) Flip the coin

2) If the coin is heads, (Heads & Monday) is selected. (This represents picking the only day SB is awake in the experiment when the coin lands heads.)

If the coin is tails, pick one of the events (Tails&Monday) , (Tails&Tuesday), giving each a probability of 1/2 of being the event selected. (This represents picking one of the events in the experiment "at random" given the coin has landed tails.)
In words, one can visualize SB's thought process upon awakening as: "I might be in either version the experiment the heads-version or the tails- version. I'll assume there is an equal chance of being in either. Given that I'm in the heads-version, it must be Monday. Given I'm in the tails version, it could be Monday or Tuesday. I'll assign an equal probability to those events."
One can object that SB is making unwarranted assumptions, but so is a "thirder" probability model. One can object that SB gets the "wrong" answers, but they are not wrong by the "halfer" probability model. The are "wrong" if the "thirder" model is assumed.

What's especially bizarre about this is that, as far as the thought experiment goes, it doesn't make any difference whether you flip the coin on Monday morning or on Tuesday morning, since it's not necessary to consult the result until Tuesday morning.

SB knows nothing definite about the day or whether the coin must have been flipped already. If you're thinking about the mental processes of someone who knows the coin has not been flipped, you're not thinking about SB's mental processes.

 

[bahamagreen]

I think Ken G #585 has got it.

After being briefed on the experiment and before being put to sleep Sunday evening rational SB wonders to herself:

"What is my present credence of heads right now, and what will be my credence of heads when I find myself future awake in an interview?"

What are thirders figuring would be her answers? If the answers are different, how is imagining being awake later different from waiting to find herself awake later, as far as calculations?

 

[stevendaryl]

"Halfer" answers will seem bizare if you don't think in terms of probability model that is consistent with "halfer" answer.

Forget about probability models. Just consider that Sleeping Beauty has just awakened. You tell her that today is Monday and that tomorrow we're going to flip a coin to decide whether to wake her up (and if you do wake her up, she'll have no memory of Monday having happened). You ask her: What's the likelihood that the coin flip tomorrow will be heads. The halfer answer has to be 2/3.

What probability model can make that sensible? How does something that we're going to do tomorrow after the coin flip affect the probability of the coin flip?

 

[stevendaryl]

I think Ken G #585 has got it.

After being briefed on the experiment and before being put to sleep Sunday evening rational SB wonders to herself:

"What is my present credence of heads right now, and what will be my credence of heads when I find myself future awake in an interview?"

What are thirders figuring would be her answers? If the answers are different, how is imagining being awake later different from waiting to find herself awake later, as far as calculations?

Suppose instead of waking once if heads and twice if tails, we said that you wake zero times if heads and two times if tails. Then it would make perfect sense for Sleeping Beauty to say:

"The probability that the coin flip will be heads is 1/2. But if on Monday or Tuesday I'm awake, I'll know that it was definitely tails."

So the number of times being awakened definitely affects her answer of "What credence of heads will you give when you're awakened?"

 

[Stephen Tashi]

The thirders are correctly responding to the question asked, which is, what is her credence given that she has been wakened.

One does not need to be "thirder" to compute what answer should be given.

Consider this situation. A coin is flipped. The bet is "You get $1 if the coin lands heads and you lose $2 if the coin lands tails". Is the fair price for the bet equal to your credence for the event "The coin landed heads"?

No, it isn't.. By the definition of credence cited several times in this thread, your credence for "The coin landed heads" is supposed to be the fair price you set for the bet "You get $1 if the coin lands heads" - with no other consequences.

 

[Stephen Tashi]

Forget about probability models.

I don't think questions about probability can be decided by forgetting about probability models - but I understand the frustration they bring.

Just consider that Sleeping Beauty has just awakened. You tell her that today is Monday and that tomorrow we're going to flip a coin to decide whether to wake her up (and if you do wake her up, she'll have no memory of Monday having happened). You ask her: What's the likelihood that the coin flip tomorrow will be heads. The halfer answer has to be 2/3.

What probability model can make that sensible?

I just gave you one.

How does something that we're going to do tomorrow after the coin flip affect the probability of the coin flip?

If the coin is flipped before SB is awakened for the first time, how do "thirders" explain that SB's awakening goes back in time and affects how the coin landed?

Conditional probabilities do not imply any physical cause-and-effect relations. Neither "halfers' nor "thirders" should expect to justify their answers by some cause-and-effect physical model.

 

[Ken G]

One does not need to be "thirder" to compute what answer should be given.

Consider this situation. A coin is flipped. The bet is "You get $1 if the coin lands heads and you lose $2 if the coin lands tails". Is the fair price for the bet equal to your credence for the event "The coin landed heads"?

Of course not, and that's why I am not using that meaning. I am saying, the credence is the fair price she would pay, X, to receive a $1 payoff if she's right. That's exactly what is 1/3 for "heads" each time she is wakened, as is easy to show in repeated trials.

In my opinion it is crucial to avoid probability arguments, because then people ask questions like "what is the actual probability the coin came out heads," and of course there is nothing that goes back in time and changes that in some absolute sense, like a force on the coin. We should instead think in terms of fair betting odds, which is what credence actually is. One can do it with probability, but subtle issues enter, like what counts as information that can cause a reassessment of a probability. But betting odds make the situation way easier, it becomes an actual way to make or lose money. We can actually do the experiment, and bilk the halfers out of their life savings.

 

[Ken G]

Suppose instead of waking once if heads and twice if tails, we said that you wake zero times if heads and two times if tails. Then it would make perfect sense for Sleeping Beauty to say:

"The probability that the coin flip will be heads is 1/2. But if on Monday or Tuesday I'm awake, I'll know that it was definitely tails."

So the number of times being awakened definitely affects her answer of "What credence of heads will you give when you're awakened?"

Yes, this example shows clearly that being wakened does indeed involve new information that changes SB's assessment of the heads probability. What is confusing the halfers is they think the heads probability is a set thing, specified when the coin is flipped, but probabilities actually mean what is consistent with the information you have. In the way the puzzle is formulated, it's subtle what that new information is, so your version makes it clear that being awakened is a form of information. For halfers reading this, if you play bridge, consider that every bridge hand has a probability of being dealt, but much of the skill of bridge amounts to updating that probability using information gathered during the bidding and play. So "the probability they have the queen" is not set by the deal, because probabilities are more active animals than that. That this is a subtle point is my reason for avoiding probabilities in favor of simply odds payoffs, ergo the relevance of the game of poker for those who have played it.

 

[Stephen Tashi]

Of course not, and that's why I am not using that meaning. I am saying, the credence is the fair price she would pay, X, to receive a $1 payoff if she's right.

But you are adding the condition that she has to pay X twice if she's wrong. So you are making the bet have consequences similar to the example I gave.

That's exactly what is 1/3 for "heads" each time she is wakened, as is easy to show in repeated trials.

I agree that saying 1/3 is the correct strategy. What I'm saying is the definition of credence for an event E is supposed to be a "pure" bet on E. Your opinion is that because the experimenter uses the words "What is your credence for the event 'the coin landed heads'" that SB is being offered a "pure" bet on that event. She is not. She is being offered the bet: " If coin lands heads you win $1 and if it lands tails you lose twice what you offered for the bet".

 

[Ken G]

But you are adding the condition that she has to pay X twice if she's wrong.

No, I'm saying that she has to pay X every time she takes a bet that is wrong. That's just how betting works.

I agree that saying 1/3 is the correct strategy.

This is the crucial point-- that's all credence means. So you are not doing anything wrong in your mathematics, and you cannot be made to lose money. You are simply not using the definition of credence correctly-- that definition is, you pay X, and lose X, every time your bet is wrong, regardless of how often that may be. If X is the ratio of the cost of a bet to the payoff, and if you break even with some given X, then X is your credence. It's not an abstract definition, it's a practical one, relevant to all betting games of chance.

 

[Stephen Tashi]

No, I'm saying that she has to pay X every time she takes a bet that is wrong. That's just how betting works.
This is the crucial point-- that's all credence means.

Your'e entitled to make-up you own definition of "credence". I'm talking about the definition that has been cited several times in this thread:

https://plato.stanford.edu/entries/imprecise-probabilities/

This boils down to the following analysis:

Your degree of belief in E is p iff p units of utility is the price at which you would buy or sell a bet that pays 1 unit of utility if E, 0 if not E.

If the experimenter phrases the question as "What is your credence that it will rain tomorrow" and the consequence of SP's answer is that she will receive $1 if the coin landed heads and lose twice what she offered for the bet if it lands tails then SB's answer should be 1/3. This is the best strategy. That answer is not her credence for the event "it will rain tomorrow". To get her credence for rain tomorrow, she is supposed to buy the the bet "you get $1 if it rains tomorrow". Her credence for rain tomorrow is not measured by the price she would pay for the bet "You get $1 if it rains tomorrow and lose twice what you paid for the bet if it doesn't".

 

[bahamagreen]

Assuming the coin is flipped between explaining the experiment and putting her to sleep...

I am asking about the Wiki version of the experiment according to which she knows what she is going to be asked when interviewed. It is only natural for her to wonder why she could not answer the future interview question immediately after understanding the experiment rather than waiting for an interview. It is also rational for her to wonder and check to see if her interview answer would be different than her answer before retiring Sunday evening. I'm asking if thirders would figure she gets two different answers.

 

[Ken G]

Your'e entitled to make-up you own definition of "credence". I'm talking about the definition that has been cited several times in this thread:

https://plato.stanford.edu/entries/imprecise-probabilities/

I am not making up my own, that is precisely the definition I am using. You pay X every time you bet, that's what it says. You get $1 when you win, that's what it says. That's what I'm saying, that's what is right. Credence never has anything to do with how many times you bet, that's why that element is never mentioned in its definition. In fact, SB is free to bet any number of times she likes, she can bet 10 times in one waking, and 5 times in another. It never affects the credence at all, because none of that will ever affect the fair odds!

 

[Ken G]

It is only natural for her to wonder why she could not answer the future interview question immediately after understanding the experiment rather than waiting for an interview.

And indeed she could, she merely needs to include the information she will have available then, compared to now. (She knows she will be wakened, that's information as proven in the example where there is never any waking for a heads.) The situation is like this. You are playing bridge, and you know there's a 50-50 chance the queen was dealt to either of your opponents. So that's your credence at the start of the hand. You know you won't have to decide which way to do the finesse until later in the hand, so you play the other suits and gather information. Let's imagine you know you will be able to ascertain which opponent has the queen based on how you play the hand (it doesn't matter how you'd actually know that, it's a hypothetical example). It is natural for you to ask what your credence is going to be after you have done that-- the point is, at the beginning of the hand, you have 50-50 credence, and at that later point in the hand, you know your credence will be certain, even as you are thinking at the start of the hand. So it makes no difference when you do the thinking, what matters is what information you know you are going to have available at the time you ascertain your credence. This is a perfectly practical consideration, it is the reason airline pilots are trained to make whatever decision will allow them to gather more information before they have to make the life-or-death decision. It's the most practical consideration there is-- credence at a given moment is based on available information at that moment.

I'm asking if thirders would figure she gets two different answers.

And the answer is yes, just like the bridge player, just like the airline pilot. To get the right credence, you don't need to have the information the whole time, you only need to know, the whole time, that you will have information when you ascertain the credence.

 

[Marana]

Use four volunteers, and the four cards I described before (with (H,Mon), (H,Tue), (T,Mon), and (T,Tue) written on them). Deal the cards to the four, and put them in separate rooms. Using one coin flip, and waken three of them on Monday, and Tuesday. Leave the one whose dealt card matches both the day, and the coin flip, asleep. Ask each for her confidence that the coin matches her card.

Obviously, if you show each Beauty her card, her answer has to be the same as the original Beauty's. Since it is the same regardless of what card is dealt, you don't have to show it to any of them. If you don't show it to any of them, you can put all three awake Beauties in a room together to discuss their answers. All have the same information, so all answers have to be the same. Since exactly one of the three has a card that matches the coin flip, that answer must be 1/3.

You seem to be making a big assumption without explicitly stating it.

Suppose there are 1001 beauties: 1 winner who wakes up 1000 days in a row, and 1000 losers who wake up once in that time. Each thinks on sunday "There is a 1000/1001 chance that I wake up next to the winner."

Your assumption is that not only must they answer the same way to "am I the winner?", but that the correct way to compute probability is to divide it up equally among those with symmetric information. That is, even if I correctly believe there is a 1000/1001 chance that I will wake up next to the winner, when I actually do wake up next to someone I should split the probability evenly, giving myself a 1/2 chance of being the winner.

But you haven't justified that assumption. I argue that, if I am correct in believing I have a 1000/1001 chance of waking up next to the winner, then when I wake up next to someone it can make me think they are the winner. I am no longer indifferent to them because they are across from me when I wake up, which I didn't know would happen. But I did know I would wake up on the same day as myself.

The fact that they have symmetric information that leads them to believe I am the winner is odd, but I don't see why it requires me to "divide the probability evenly."

The bet is implied by the definition of credence. So it is every time she is asked about her credence, which is every interview.

They were stated in the original problem. Beauty is asked about her credence in each interview so the implied bet is necessarily offered each interview also.

This is just false. The definition of credence does not provide the ability to see into the future or the forgotten past.

You are adding that ability because credence is extremely hard to define in the actual problem. But sleeping beauty considering a bet being offered now does not imply that she can rely on the existence of specific past or future bets. That simply isn't in the definition, and it has nothing to do with cheating on the part of the experimenters.

It may be that we can't come up with a coherent way to apply the definition of credence to this situation. Or maybe we can say that she accepts bets on sunday when she has P(H) = 1/2. Or maybe we can use reflection to wednesday at noon when she has P(H) = 1/2. Or maybe we should consider "surprise", in which case the lottery example is compelling to me. The thirder answer means that sleeping beauty can become arbitrarily confident that she won an arbitrarily unlikely lottery.

A is Beauty is awakened during the experiment (i.e. with amnesia, being interviewed, and being asked her credence that it is heads).

Are you under the impression that beauty always has amnesia when asked her credence? The experiment goes like this:

1/2: sunday coin flip heads -> monday interview
1/2: sunday coin flip tails -> monday interview -> (amnesia regarding monday) -> tuesday interview

Note that there has been no amnesia before the monday interview, and that there is never any amnesia that severs the causal link between the sunday coin flip and the current interview.

This is a crucial point. It is not enough for sleeping beauty to be told the rules when she wakes up, even if you tell her it is the first time the experiment has ever been performed. If she has total amnesia, then it is too late. Her awakening is already selected by the time you explain, and it becomes a different problem. The unbroken causal link from sunday is essential.