“ADDED: This is depressing. Here we have a collection of people who have studied probability problems and anthropic reasoning and all the relevant issues for years. And we have a question that is, on the scale of questions in the project of preparing for AGI, a small, simple one. It isn’t a tricky semantic or philosophical issue; it actually has an answer. And the LW community is doing worse than random at it.”
That’s why I posted this to begin with. It is interesting that we can’t come to an agreement on the solution to this problem, even though it involves very straightforward probability. Heck, I got heavily down voted after making statements that were correct. People are getting thrown off by doing the wrong kind of frequency counting.
--
However, I should note that the event ‘sleeping beauty is awake’ is equivalent to ‘sleeping beauty has been woken up at least once’ because of the amnesia. The forgetfulness aspect of the problem is why the solution is 1⁄2.
Distressingly few people have publicly changed their mind on this thread. Various people show great persistence in believing the wrong answer—even when the problem has been explained. Perhaps overconfidence is involved.
I changed my mind from “1/3 is the right answer” to “The answer is obviously 1⁄2 or 1⁄3 once you’ve gotten clear on what question is being asked”. I’m not sure if I did so publicly. It seems to me that other folks have changed their minds similarly. I think I see an isomorphism to POAT here, as well as any classic Internet debate amongst intelligent people.
I’m also not sure if you’re serious, but if you assign a 50% probability to the relevant question being the one with the correct answer of ‘1/2’ and a 50% probability to the relevant question being the one with the correct answer of ‘1/3’ then ‘5/12’ should maximize your payoff over multiple such cases if you’re well-calibrated.
Phil and I seem to think the problem is sufficiently clearly specified to give an answer to. If you think 1⁄2 is a defensible answer, how would you reply to Robin Hanson’s comment?
FWIW, on POAT I am inclined towards “Whoever asked this question is an idiot”.
there are some problems similar to this one for which the answer is 1⁄2
there are some problems similar to this one for which the answer is 1⁄3
people seem to be disagreeing which sort of problem this is
all debate has devolved to debate over the meanings of words (in the problem statement and elsewhere)
Given this, I think it’s obvious that the problem is ambiguous, and arguing whether the problem is ambiguous is counterproductive as compared to just sorting out which sort of problem you’re responding to and what the right answer is.
IMHO, different people giving different answers to problems does not mean it is ambiguous. Nor does people disagreeing over the meanings of words. Words do have commonly-accepted meanings—that is how people communicate.
I’m coming around to the 1⁄2 point of view, from an initial intuition that 1⁄3 made most sense, but that it mostly depended on what you took “credence” to mean.
My main new insight is that the description of the set-up deliberately introduces confusion, it makes it seem as if there are two very different situations of “background knowledge”, X being “a coin flip” and X’ being “a coin flip plus drugs and amnesia”. So that P(heads|X) may not equal P(heads|X’).
This comment makes the strongest case I’ve seen that the difference is one that makes no difference. Yes, the setup description strongly steers us in the direction of taking “credence” to refer to the number of times my guess about the event is right. If Beauty got a candy bar each time she guessed right she’d want to guess tails. But on reflection what seems to matter in terms of being well-calibrated on the original question is how many distinct events I’m right about.
Take away the drug and amnesia, and suppose instead that Beauty is just absent-minded. On Tuesday when you ask her, she says: “Oh crap, you asked me that yesterday, and I said 1⁄2. But I totally forget if you were going to ask me twice on tails or on heads. You’d think with all they wrote about this setup I’d remember it. I’ve no idea really, I’ll have to go with 1⁄2 again. Should be 1 for one or the other, but what can I say, I just forget.”
I’m less than impressed with the signal-to-noise ratio in the recent discussion, in particular the back-and-forth between neq1 and timtyler. As a general observation backed by experience in other fora, the more people are responding in real time to a controversial topic, the less likely they are to be contributing useful insights.
I’ve been thinking 1⁄2 as well (though I’m also definitely in the “problem is underdefined” camp).
Here is how describe the appropriate payoff scheme. Prior to the experiment (but after learning the details) Beauty makes a wager with the Prince. If the coin comes up heads the Prince will pay Beauty $10. If it comes up tails Beauty will pay $10. Even odds. This wager represents Beauty’s prior belief that the coin is fair and head/tails have equal probability: her credence that heads will or did come up. At any point before Beauty learns what day of the week it is she is free alter the bet such that she takes tails but must pay $10 more dollars to do so (making the odds 2:1).
Beauty should at no point (before learning what day of the week it is) alter the wager. Which means when she is asked what her credence is that the coin came up heads she should continue to say 1⁄2.
This seems at least as good an payoff interpretation as a new bet every time Beauty is asked about her credence.
You don’t measure an agent’s subjective probability like that, though—not least because in many cases it would be bad experimental methodology. Bets made which are intended to represent the subject’s probability at a particular moment should pay out—and not be totally ignored. Otherwise there may not be any motivation for the subject making the bet to give an answer that represents what they really think. If the subject knows that they won’t get paid on a particular bet, that can easily defeat the purpose of offering them a bet in the first place.
If Beauty forgets what is going on—or can’t add up—her subjective probability could potentially be all over the shop.
However, the problem description states explicitly that: “During the experiment, she has no access to anything that would give a clue as to the day of the week. However, she knows all the details of the experiment.”
This seems to me to weigh pretty heavily against the hypothesis that she may have forgotten the details of the experiment.
In the case where she remembers what’s going on, when you ask her on Tuesday what her credence is in Heads, she says “Well, since you asked me yesterday, the coin must have come up Tails; therefore I’m updating my credence in Heads to 0.”
The setup makes her absent-minded (in a different way than I suggest above). It erases information she would normally have. If you told her “It’s Monday”, she’d say 1⁄2. If you told her “It’s Tuesday”, she’d say 0. The amnesia prevents Beauty from conditioning on what day it is when she’s asked.
Prior to the experiment, Beauty has credence 1⁄2 in either Heads or Tails. To argue that she updates that credence to 1⁄3, she must be be taking into account some new information, but we’ve established that it can’t be the day, as that gets erased. So what it is?
Jonathan_Lee’s post suggests that Beauty is “conditioning on observers”. I don’t really understand what that means. The first analogy he makes is to an identical-copy experiment, but we’ve been over that already, and I’ve come to the conclusion that the answer in that case is “it depends”.
Yes. I noted then that the description of the setup could make a difference, in that it represents different background knowledge.
It does not follow that it does make a a difference.
When I say “prior to the experiment”, I mean chronologically, i.e. if you ask Beauty on Sunday, what her credence is then in the proposition “the coin will come up heads”, she will answer 1⁄2.
Once Beauty wakes up and is asked the question, she conditions on the fact that the experiment is now ongoing. But what information does that bring, exactly?
When Beauty knows she will be the subject of the experiment (and its design), she will know she is more likely to be observing tails. Since the experiment involves administering Beauty drugs, it seems fairly likely that she knew she would be the subject of the experiment before it started—and so she is likely to have updated her expectations of observing heads back then.
If she is asked: “if you wake up with amnesia in this experiment, what odds of the coin being heads will you give”, then yes. She doesn’t learn anything to make her change her mind about the odds she will give after the experiment has started.
That isn’t a symmetrical question. We’re not asking for her belief about what odds she will give. We’re asking what her odds are for a particular event (namely a coin flip at time t1 being heads).
The question “What is your credence now for the proposition that our coin landed heads?” doesn’t appear to make very much sense before the coin is flipped. Remember that we are told in the description that the coin is only flipped once—and that it happens after Beauty is given a drug that sends her to sleep.
Beauty should probably clarify with the experimenters which previous coin is being discussed, and then, based on what she is told about the circumstances surrounding that coin flip, she should use her priors to answer.
The English language doesn’t have a timeless tense. So we can’t actually phrase the question without putting the speaker into some time relative to the event we’re speaking of. But that doesn’t mean we can’t recognize that the question being asked is a timeless one. We have a coordinate system that lets us refer to objects and events throughout space and time… it doesn’t matter when the agent is: the probability of the event occurring can be estimated before, after and during just as easily (easy mathematically, not practically). That is why I used the phrasing “the coin flip at time t1 being heads”. The coin flip at t1 can be heads or tails. Since we know it is a fair coin toss we start with P=1/2 for heads. If you want the final answer to be something other than 1⁄2 you need to show when and how Beauty gets additional information about the coin toss.
The question asked in the actual problem has the word “now” in it. You said I didn’t answer a “symmetrical” question—but it seems as though the question you wanted me to answer is not very “symmetrical” either.
If Beauty is asked before the experiment the probabality she expects the coin to show heads at the end of the experiment, she will answer 1⁄2. However, in the actual problem she is not asked that.
We’re supposed to be Bayesians. It doesn’t matter whether the question asks “now” “in 500 B.C.E.” or “at the heat death of the universe” unless our information has changed, the time the prediction is made is irrelevant.
(ETA: Okay, I guess at the heat death of the universe the information would have changed. But you get my point :-)
But here you’ve put the time-indexical “now” into your description of the event. You’re asking for P(it is night, now). In the Beauty case question asked is what is P(heads), now. In the first case every moment that goes by we’re talking about a temporally distinct event. You’re actually asking about a different event every moment- so it isn’t surprising that the answer changes from moment to moment. The Sleeping Beauty problem is always about the same event.
The coin flip doesn’t change—but Beauty does. She goes in one end of the expertiment and comes out the other side, and she knows roughly where she is on that timeline. Probabalities are subjective—and in this example we are asked for Beauty’s “credence”—i.e. her subjective probability at a particular point in time. That’s a function of the observer, not just the observed.
Yes. But subjective probability is a function of the information someone has not where they are on the time-line. Which is why people keep asking what information Beauty is updating on. We’re covering 101 stuff at this point.
...and going round in circles, I might note. We did already discuss the issue of exactly when Beauty updates close by—here.
Also, we already know where we differ. We consider “subjective probability” to refer to different things. Given your notion of “subjective probability”, your position makes perfect sense, IMO. I just don’t think that is how scientists generally use the term.
Well you tried to answer the question. I suggested your answer was ridiculous and explained why and I have been rebutting your responses since then. So no, we’re not going in circles. I’m objecting to your answer to the updating question and rebutting your responses to my objection.
Here is what happened in this thread.
You suggested that Beauty would have estimated heads at 1⁄3 prior to the experiment.
I said ‘Wha?!?’
You tried to make Beauty’s pre-experiment estimation about what she was going to say when she woke up.
I pointed out that that question was about a different event (the saying) than the question “What is your credence now for the proposition that our coin landed heads?” is about (the coin flip)
You claimed that it didn’t make sense to ask that question (about the coin having landed heads) before the coin flip happens.
I showed how even though English requires us to use tense we can make the question time symmetrical by inventing temporal coordinates (t1) and speaking of subjective probability of heads at t1 at any time Beauty exists.
You claimed that the probability of heads at the end of the experiment was somehow different from the probability of heads at some other time (presumably when she is asked).
I pointed out that time is irrelevant and what matters is her information- an elementary point which I shouldn’t have to make to someone who was last night trashing the OP for supposedly not knowing anything about probability (and I’m a philosopher not a math guy!).
In conclusion: My claim is that for Beauty to answer 1⁄3 for the probability of the time invariant event “coin toss by experimenter at time t1 being heads” she needs to get new information since the prior for that even is obviously 1⁄2. No one has ever pointed to what new information she gets. You tried to claim that Beauty updates as soon as she gets the details of the experiment: but that can’t be right. The details of the experiment can’t alter the outcome of a fair coin toss. So where is the updating?!
It’s hard to tell but I’m not sure your notion of “subjective probability” is coherent- specifically because you keep talking about different events depending on what time you’re in. That sounds like a recipe for disaster. But alright.
I just don’t think that is how scientists generally use the term.
Does this mean we can just agree to specify payouts in our probability problems from now on? Or must we now investigate which one of us is using the term the way scientists do? Unfortunately this disagreement suggest to me that scientists may not know exactly what they mean by subjective probability.
Subjective probability is a basic concept in decision theory. Scientists have certainly tried hard to say exactly what they mean by the term. E.g. see this one, from 1963:
“A Definition of Subjective Probability”—F. J. Anscombe; R. J. Aumann
Sure. I don’t see anything in there to suggest that subjective probability isn’t time symmetrical (by which I mean that a subjective probability regarding an event can be held at any time and there is not reason for the probability to change unless the person’s evidence changes). Can you do a better job formalizing what your alternative is?
However, I should not that the event ‘sleeping beauty is awake’ is equivalent to ‘sleeping beauty has been woken up at least once’ because of the amnesia.
My assumptions and use of probability laws are clearly stated above. Tell me where I made a mistake, otherwise just saying “you’re wrong” is not going to move things forward.
“Suppose this experiment were repeated 1,000 times. We would expect to get 500 heads and 500 tails. So Beauty would be awoken 500 times after heads on Monday, 500 times after tails on Monday, and 500 times after tails on Tuesday. In other words, only in a third of the cases would heads precede her awakening. So the right answer for her to give is 1⁄3. This is the correct answer from Beauty’s perspective.”
That gives:
P(monday and heads)=500/1500. P(monday and tails)=500/1500. P(tuesday and tails)=500/1500.
You appear to have gone wrong by giving a different answer—based on a misinterpretation of the meaning of the interview question, it appears.
So you are not willing to tell me where I made a mistake?
P(heads)=1/2, p(monday | heads)=1. Which one of these is wrong?
You’re using expected frequencies to estimate a probability, apparently. But you’re counting the wrong thing. What you are calling P(monday and heads) is not that. There is a problem with your denominator. Think about it. Your numerator has a maximum value of 1000 (if the experiment was repeated 1000 times). Your denominator has a maximum value of 2000. If the maximum possible values of the numerator and denominator do not match, there is a problem. You have an outcome-dependent denominator. Try taking expectation of that. You won’t get what you think you’ll get.
Re: “If the maximum possible values of the numerator and denominator do not match, there is a problem.
The total possible number of awakenings is 2000.
That represents all tails—e.g.:
P(monday and heads) = 0/2000;
P(monday and tails) = 1000/2000;
P(tuesday and tails) = 1000/2000;
These values add up to 1 - i.e. the total numerators add up to the commonn denominator. That is the actual constraint. The maximum possible value of the numerator in each individual fraction is permitted to be smaller than the common denominator—that is not indicative of a problem.
Oh, it is a huge problem. It proves that your ratio isn’t of the form # of events divided by # of trials. Your ratio is something else. The burden is on you to prove that it actually converges to a probability as the number of trials goes to infinity.
Using cell counts and taking a ratio leads to a probability as the number of trials goes to infinity if you have independent draws. You don’t. You have a strange dependence in there that messes things up. Standard theory doesn’t hold. Your thing there is estimating something, you just don’t know what it is
The total number of events (statements by Beauty) adds up to the total number of trials (interviews).
You should not expect the number of statements by beauty on Monday to add up to the total number of interviews alltogether. It adds up to the number of interviews on Monday. This is not very complicated.
Do you have to make a condescending remark every time you respond? You told me things that I already know, and then said “This is not very complicated.” Great, but nothing accomplished.
You are using an estimator that is valid when you have counts from independent trials. Coin flips are independent here, but interviews are not. You need to take that into account.
It is the plain truth. I don’t know why you are asking such silly questions in public. Maybe you have a weak background in this sort of maths. Or maybe you just don’t like admitting that you posted a whole bunch of inaccurate nonsense—and so keep digging yourself deeper in.
You show no sign of being able to understand your problems—so it seems to me as though there is little point in continuing to point them out. You can’t say I didn’t try to help you sort yourself out.
Well, I have a phd in biostatistics and teach Bayesian data analysis at the University of Pennsylvania, so I either have background in such matters or Penn isn’t real careful on who they hire.
The fact that I am very careful about these kinds of problems is what lead me to discover the flaw in the 1⁄3 argument—it wasn’t obvious to me at first.
Hi, Nancy! I haven’t researched this issue. I imagine the results would depend on the details of the situation, the relative status of the participants, etc. I recommend you consult a social psychologist—if you are sincerely looking for answers.
Uh—improving neq1′s state of knowledge was not the intended purpose of that post.
I have already written literally dozens of posts attempting to improving the state of knowledge of other participants on this thread. That post was publicly explaining why I am now likely to stop—just so there is no subsequent confusion about the issue.
Right—but I call a spade a spade, don’t beat about the bush, say what I think—etc.
Insulating others from what I think in order to protect their egos is not my style. If I did that people would always be wondering if I meant what I said—or whether I was shielding them from my true opinions in order to protect their egos. In the long run, it is best to just speak the truth, as I see it, IMO. At least then, others know where I stand.
There are a lot of approaches one can take when interacting with other people. Your approach leads me to not want to make your acquaintance. The same isn’t true for most of the other people here, even the ones who disagree with me.
In the spirit of experimentation, I’m going to try giving up being oblique.
My primary motivation is not to do you a favor. My purpose is also not to protect neq1.
It is to convey that the purpose of this website is to work on thinking clearly, to some extent to further the creation of FAI, and also to improve skills at living. There’s also a little pleasant socializing.
However, insulting people (or having an atmosphere where insults are accepted) does not further any of the purposes of the site.
I believe that people generally think less well when they’ve been insulted. Also, I’ve been online for a long time. Insults are pretty much similar to each other—in other words, they’re noise, not signal so far as anything about the world generally is concerned.. They’re signal about emotional state and/or attempted dominance, but (as should be clear from the conversation so far), not a terribly clear signal.
What’s worse, insults are likely to lead to more insults.
I’m not a moderator, but I’m asking you not to dump hostility here.
Nancy, your beliefs about the average effect of insults on people do not seem to me to be a good reason to avoid bluntly telling people when they are behaving badly. IMO, you are not properly considering the positive effects of pointing out such bad behaviour. If someone behaves badly, and you don’t tell them, they don’t learn. Others might think their behaviour is acceptable. Still others might think you approve of their behaviour—and so on. It is not as though I had not tried all manner of rational argument first. Yes, people might be insulted or offended by someone else pointing out what is going on—if it reflects badly on them, but that is—ultimately—their business.
Not especially—and certainly not from my point of view. Alas, I found responding to your comments to be a waste of my time and energy. Especially so with your “oblique” comments. So, overall, I would rather you had not bothered commenting in the first place.
In the spirit of experimentation, I’m going to try giving up being oblique.
My primary motivation is not to do you a favor. My purpose is also not to protect neq1.
It is to convey that the purpose of this website is to work on thinking clearly, to some extent to further the creation of FAI, and also to improve skills at living. There’s also a little pleasant socializing.
However, insulting people (or having an atmosphere where insults are accepted) does not further any of the purposes of the site.
I believe that people generally think less well when they’ve been insulted. Also, I’ve been online for a long time. Insults are pretty much similar to each other—in other words, they’re noise, not signal so far as anything about the world generally is concerned.. They’re signal about emotional state and/or attempted dominance, but (as should be clear from the conversation so far), not a terribly clear signal.
What’s worse, insults are likely to lead to more insults.
I’m not a moderator, but I’m asking you not to dump hostility here.
In the spirit of experimentation, I’m going to try giving up being oblique.
My primary motivation is not to do you a favor. My purpose is also not to protect neq1.
It is to convey that the purpose of this website is to work on thinking clearly, to some extent to further the creation of FAI, and also to improve skills at living. There’s also a little pleasant socializing.
However, insulting people (or having an atmosphere where insults are accepted) does not further any of the purposes of the site.
I believe that people generally think less well when they’ve been insulted. Also, I’ve been online for a long time. Insults are pretty much similar to each other—in other words, they’re noise, not signal so far as anything about the world generally is concerned.. They’re signal about emotional state and/or attempted dominance, but (as should be clear from the conversation so far), not a terribly clear signal.
What’s worse, insults are likely to lead to more insults.
I’m not a moderator, but I’m asking you not to dump hostility here.
In the spirit of experimentation, I’m going to try giving up being oblique.
My primary motivation is not to do you a favor. My purpose is also not to protect neq1.
It is to convey that the purpose of this website is to work on thinking clearly, to some extent to further the creation of FAI, and also to improve skills at living. There’s also a little pleasant socializing.
However, insulting people (or having an atmosphere where insults are accepted) does not further any of the purposes of the site.
I believe that people generally think less well when they’ve been insulted. Also, I’ve been online for a long time. Insults are pretty much similar to each other—in other words, they’re noise, not signal so far as anything about the world generally is concerned.. They’re signal about emotional state and/or attempted dominance, but (as should be clear from the conversation so far), not a terribly clear signal.
What’s worse, insults are likely to lead to more insults.
I’m not a moderator, but I’m asking you not to dump hostility here.
But there are probably an infinite number of propositions that you actually believe, and even an infinite number of relevant propositions that you actually believe. You choose which things that you think to actually say (I’m just assuming that everything you think ‘out loud’ doesn’t get posted to Less Wrong, since I assume you have more thoughts than I’ve observed comments from you). As long as you’re leaving out an infinite amount of information, you might as well also leave out insulting language.
I said he was asking “silly questions”. However, that is true—and was not “insulting language”. If you think I was using “insulting language”, you will have to be more specific about what you mean.
As to the possibility of you making a more general point, IMO, systematically not speaking truths that might cause offense would have bad results—especially for truth-seekers:
“Unfortunately some people take offense more easily than others. Also, some people are offended by true statements.”
It is the same with pressuring other people to not speak truths that might cause offense. That too, would have—and has had—seriously unpleasant long-term effects.
That was implied information: “it seems to me as though there is little point in continuing to point them out. You can’t say I didn’t try to help you sort yourself out.”
“Likely to stop” is a probabalistic statement. I am still likely to stop posting on this thread soon. I have done my bit to promote the correct answer to this problem. A top level post explains the correct answer in some detail. I feel as though my work here is done.
Were someone else exhibiting similar posting behavior, would you draw the same conclusion? You may sincerely desire to terminate your conversation with neq1, but you appear to cesire* to continue it.
* A “cesire” is a motivator to action that works like a desire even when accompanied by a conflicting desire—much like an alief can induce emotional reactions in the same way as beliefs even in the presence of a contrary belief.
The conclusion that you feel your work is done. Such a state removes the desire to continue responding to neq1, and—as such a desire is the only apparent reason to respond to neq1 - leads to a cessation of posts in the associated thread(s). This has not occurred.
I haven’t argued about the topic of this post for a little while now—and certainly not since writing “I feel as though my work here is done”.
Rather I am here defending my reputation against assaults from people who don’t like my posting style—and seem keen to let everyone else know of their disapproval. I’ll probably give up with that too, soon enough.
Were someone else exhibiting similar posting behavior, would you draw the same conclusion? You may sincerely desire to terminate your conversation with neq1, but you appear to cesire* to continue it.
* A “cesire” is a motivator to action that works like a desire even when accompanied by a conflicting desire—much like an alief can induce emotional reactions in the same way as beliefs even in the presence of a contrary belief.
Actually, I have supplied my real name (in a previous post I linked to my blog, which has my name). I’m confident my colleagues would be in agreement with me.
For Jack’s bookie, I agree, you have to use 1⁄3 – but if you want to calculate a distribution on how much cash Beauty has after the experiment given different betting behavior, it no longer works to treat Monday and Tuesday as mutually exclusive.
“ADDED: This is depressing. Here we have a collection of people who have studied probability problems and anthropic reasoning and all the relevant issues for years. And we have a question that is, on the scale of questions in the project of preparing for AGI, a small, simple one. It isn’t a tricky semantic or philosophical issue; it actually has an answer. And the LW community is doing worse than random at it.”
That’s why I posted this to begin with. It is interesting that we can’t come to an agreement on the solution to this problem, even though it involves very straightforward probability. Heck, I got heavily down voted after making statements that were correct. People are getting thrown off by doing the wrong kind of frequency counting.
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However, I should note that the event ‘sleeping beauty is awake’ is equivalent to ‘sleeping beauty has been woken up at least once’ because of the amnesia. The forgetfulness aspect of the problem is why the solution is 1⁄2.
I’d like to see a model of how a group of people is supposed to improve their initial distribution of beliefs in a problem with a true/false answer.
Distressingly few people have publicly changed their mind on this thread. Various people show great persistence in believing the wrong answer—even when the problem has been explained. Perhaps overconfidence is involved.
I changed my mind from “1/3 is the right answer” to “The answer is obviously 1⁄2 or 1⁄3 once you’ve gotten clear on what question is being asked”. I’m not sure if I did so publicly. It seems to me that other folks have changed their minds similarly. I think I see an isomorphism to POAT here, as well as any classic Internet debate amongst intelligent people.
I’m not sure whether this is legitimate or a joke, but if the question is unclear about whether 1⁄2 or 1⁄3 is better, maybe 5⁄12 is a good answer.
I’m also not sure if you’re serious, but if you assign a 50% probability to the relevant question being the one with the correct answer of ‘1/2’ and a 50% probability to the relevant question being the one with the correct answer of ‘1/3’ then ‘5/12’ should maximize your payoff over multiple such cases if you’re well-calibrated.
Phil and I seem to think the problem is sufficiently clearly specified to give an answer to. If you think 1⁄2 is a defensible answer, how would you reply to Robin Hanson’s comment?
FWIW, on POAT I am inclined towards “Whoever asked this question is an idiot”.
Actually I think it would make more sense to reply to my own comment in response to this. link
I am not sure that is going anywhere.
Personally, I think I pretty-much nailed what was wrong with the claim that the problem was ambiguous here.
I think that we’ve established the following:
there are some problems similar to this one for which the answer is 1⁄2
there are some problems similar to this one for which the answer is 1⁄3
people seem to be disagreeing which sort of problem this is
all debate has devolved to debate over the meanings of words (in the problem statement and elsewhere)
Given this, I think it’s obvious that the problem is ambiguous, and arguing whether the problem is ambiguous is counterproductive as compared to just sorting out which sort of problem you’re responding to and what the right answer is.
IMHO, different people giving different answers to problems does not mean it is ambiguous. Nor does people disagreeing over the meanings of words. Words do have commonly-accepted meanings—that is how people communicate.
I’m coming around to the 1⁄2 point of view, from an initial intuition that 1⁄3 made most sense, but that it mostly depended on what you took “credence” to mean.
My main new insight is that the description of the set-up deliberately introduces confusion, it makes it seem as if there are two very different situations of “background knowledge”, X being “a coin flip” and X’ being “a coin flip plus drugs and amnesia”. So that P(heads|X) may not equal P(heads|X’).
This comment makes the strongest case I’ve seen that the difference is one that makes no difference. Yes, the setup description strongly steers us in the direction of taking “credence” to refer to the number of times my guess about the event is right. If Beauty got a candy bar each time she guessed right she’d want to guess tails. But on reflection what seems to matter in terms of being well-calibrated on the original question is how many distinct events I’m right about.
Take away the drug and amnesia, and suppose instead that Beauty is just absent-minded. On Tuesday when you ask her, she says: “Oh crap, you asked me that yesterday, and I said 1⁄2. But I totally forget if you were going to ask me twice on tails or on heads. You’d think with all they wrote about this setup I’d remember it. I’ve no idea really, I’ll have to go with 1⁄2 again. Should be 1 for one or the other, but what can I say, I just forget.”
I’m less than impressed with the signal-to-noise ratio in the recent discussion, in particular the back-and-forth between neq1 and timtyler. As a general observation backed by experience in other fora, the more people are responding in real time to a controversial topic, the less likely they are to be contributing useful insights.
I’m not ruling out changing my mind again. :)
I’ve been thinking 1⁄2 as well (though I’m also definitely in the “problem is underdefined” camp).
Here is how describe the appropriate payoff scheme. Prior to the experiment (but after learning the details) Beauty makes a wager with the Prince. If the coin comes up heads the Prince will pay Beauty $10. If it comes up tails Beauty will pay $10. Even odds. This wager represents Beauty’s prior belief that the coin is fair and head/tails have equal probability: her credence that heads will or did come up. At any point before Beauty learns what day of the week it is she is free alter the bet such that she takes tails but must pay $10 more dollars to do so (making the odds 2:1).
Beauty should at no point (before learning what day of the week it is) alter the wager. Which means when she is asked what her credence is that the coin came up heads she should continue to say 1⁄2.
This seems at least as good an payoff interpretation as a new bet every time Beauty is asked about her credence.
You don’t measure an agent’s subjective probability like that, though—not least because in many cases it would be bad experimental methodology. Bets made which are intended to represent the subject’s probability at a particular moment should pay out—and not be totally ignored. Otherwise there may not be any motivation for the subject making the bet to give an answer that represents what they really think. If the subject knows that they won’t get paid on a particular bet, that can easily defeat the purpose of offering them a bet in the first place.
This doesn’t make any sense to me. Or at least the sense it does make doesn’t sound like sufficient reason to reject the interpretation.
If Beauty forgets what is going on—or can’t add up—her subjective probability could potentially be all over the shop.
However, the problem description states explicitly that: “During the experiment, she has no access to anything that would give a clue as to the day of the week. However, she knows all the details of the experiment.”
This seems to me to weigh pretty heavily against the hypothesis that she may have forgotten the details of the experiment.
In the case where she remembers what’s going on, when you ask her on Tuesday what her credence is in Heads, she says “Well, since you asked me yesterday, the coin must have come up Tails; therefore I’m updating my credence in Heads to 0.”
The setup makes her absent-minded (in a different way than I suggest above). It erases information she would normally have. If you told her “It’s Monday”, she’d say 1⁄2. If you told her “It’s Tuesday”, she’d say 0. The amnesia prevents Beauty from conditioning on what day it is when she’s asked.
Prior to the experiment, Beauty has credence 1⁄2 in either Heads or Tails. To argue that she updates that credence to 1⁄3, she must be be taking into account some new information, but we’ve established that it can’t be the day, as that gets erased. So what it is?
Jonathan_Lee’s post suggests that Beauty is “conditioning on observers”. I don’t really understand what that means. The first analogy he makes is to an identical-copy experiment, but we’ve been over that already, and I’ve come to the conclusion that the answer in that case is “it depends”.
Re: “Prior to the experiment, Beauty has credence 1⁄2 in either Heads or Tails.”
IMO, we’ve been over that adequately here. Your comment there seemed to indicate that you understood exactly when Beauty updates.
Yes. I noted then that the description of the setup could make a difference, in that it represents different background knowledge.
It does not follow that it does make a a difference.
When I say “prior to the experiment”, I mean chronologically, i.e. if you ask Beauty on Sunday, what her credence is then in the proposition “the coin will come up heads”, she will answer 1⁄2.
Once Beauty wakes up and is asked the question, she conditions on the fact that the experiment is now ongoing. But what information does that bring, exactly?
When Beauty knows she will be the subject of the experiment (and its design), she will know she is more likely to be observing tails. Since the experiment involves administering Beauty drugs, it seems fairly likely that she knew she would be the subject of the experiment before it started—and so she is likely to have updated her expectations of observing heads back then.
The question is
Your claim is that Beauty answers “1/3” before the experiment even begins?
(?!?!!)
If she is asked: “if you wake up with amnesia in this experiment, what odds of the coin being heads will you give”, then yes. She doesn’t learn anything to make her change her mind about the odds she will give after the experiment has started.
That isn’t a symmetrical question. We’re not asking for her belief about what odds she will give. We’re asking what her odds are for a particular event (namely a coin flip at time t1 being heads).
The question “What is your credence now for the proposition that our coin landed heads?” doesn’t appear to make very much sense before the coin is flipped. Remember that we are told in the description that the coin is only flipped once—and that it happens after Beauty is given a drug that sends her to sleep.
Beauty should probably clarify with the experimenters which previous coin is being discussed, and then, based on what she is told about the circumstances surrounding that coin flip, she should use her priors to answer.
The English language doesn’t have a timeless tense. So we can’t actually phrase the question without putting the speaker into some time relative to the event we’re speaking of. But that doesn’t mean we can’t recognize that the question being asked is a timeless one. We have a coordinate system that lets us refer to objects and events throughout space and time… it doesn’t matter when the agent is: the probability of the event occurring can be estimated before, after and during just as easily (easy mathematically, not practically). That is why I used the phrasing “the coin flip at time t1 being heads”. The coin flip at t1 can be heads or tails. Since we know it is a fair coin toss we start with P=1/2 for heads. If you want the final answer to be something other than 1⁄2 you need to show when and how Beauty gets additional information about the coin toss.
The question asked in the actual problem has the word “now” in it. You said I didn’t answer a “symmetrical” question—but it seems as though the question you wanted me to answer is not very “symmetrical” either.
If Beauty is asked before the experiment the probabality she expects the coin to show heads at the end of the experiment, she will answer 1⁄2. However, in the actual problem she is not asked that.
We’re supposed to be Bayesians. It doesn’t matter whether the question asks “now” “in 500 B.C.E.” or “at the heat death of the universe” unless our information has changed, the time the prediction is made is irrelevant.
(ETA: Okay, I guess at the heat death of the universe the information would have changed. But you get my point :-)
If you are locked in a lead-lined box, the answer to question “is it night time outside now” varies over time—even though you learn nothing new.
Similarly with Beauty, as she moves through the experimental procedure.
But here you’ve put the time-indexical “now” into your description of the event. You’re asking for P(it is night, now). In the Beauty case question asked is what is P(heads), now. In the first case every moment that goes by we’re talking about a temporally distinct event. You’re actually asking about a different event every moment- so it isn’t surprising that the answer changes from moment to moment. The Sleeping Beauty problem is always about the same event.
The coin flip doesn’t change—but Beauty does. She goes in one end of the expertiment and comes out the other side, and she knows roughly where she is on that timeline. Probabalities are subjective—and in this example we are asked for Beauty’s “credence”—i.e. her subjective probability at a particular point in time. That’s a function of the observer, not just the observed.
Yes. But subjective probability is a function of the information someone has not where they are on the time-line. Which is why people keep asking what information Beauty is updating on. We’re covering 101 stuff at this point.
...and going round in circles, I might note. We did already discuss the issue of exactly when Beauty updates close by—here.
Also, we already know where we differ. We consider “subjective probability” to refer to different things. Given your notion of “subjective probability”, your position makes perfect sense, IMO. I just don’t think that is how scientists generally use the term.
Well you tried to answer the question. I suggested your answer was ridiculous and explained why and I have been rebutting your responses since then. So no, we’re not going in circles. I’m objecting to your answer to the updating question and rebutting your responses to my objection.
Here is what happened in this thread.
You suggested that Beauty would have estimated heads at 1⁄3 prior to the experiment.
I said ‘Wha?!?’
You tried to make Beauty’s pre-experiment estimation about what she was going to say when she woke up.
I pointed out that that question was about a different event (the saying) than the question “What is your credence now for the proposition that our coin landed heads?” is about (the coin flip)
You claimed that it didn’t make sense to ask that question (about the coin having landed heads) before the coin flip happens.
I showed how even though English requires us to use tense we can make the question time symmetrical by inventing temporal coordinates (t1) and speaking of subjective probability of heads at t1 at any time Beauty exists.
You claimed that the probability of heads at the end of the experiment was somehow different from the probability of heads at some other time (presumably when she is asked).
I pointed out that time is irrelevant and what matters is her information- an elementary point which I shouldn’t have to make to someone who was last night trashing the OP for supposedly not knowing anything about probability (and I’m a philosopher not a math guy!).
In conclusion: My claim is that for Beauty to answer 1⁄3 for the probability of the time invariant event “coin toss by experimenter at time t1 being heads” she needs to get new information since the prior for that even is obviously 1⁄2. No one has ever pointed to what new information she gets. You tried to claim that Beauty updates as soon as she gets the details of the experiment: but that can’t be right. The details of the experiment can’t alter the outcome of a fair coin toss. So where is the updating?!
It’s hard to tell but I’m not sure your notion of “subjective probability” is coherent- specifically because you keep talking about different events depending on what time you’re in. That sounds like a recipe for disaster. But alright.
Does this mean we can just agree to specify payouts in our probability problems from now on? Or must we now investigate which one of us is using the term the way scientists do? Unfortunately this disagreement suggest to me that scientists may not know exactly what they mean by subjective probability.
Subjective probability is a basic concept in decision theory. Scientists have certainly tried hard to say exactly what they mean by the term. E.g. see this one, from 1963:
“A Definition of Subjective Probability”—F. J. Anscombe; R. J. Aumann
http://www.econ.ucsb.edu/~tedb/Courses/GraduateTheoryUCSB/anscombeaumann.pdf
Sure. I don’t see anything in there to suggest that subjective probability isn’t time symmetrical (by which I mean that a subjective probability regarding an event can be held at any time and there is not reason for the probability to change unless the person’s evidence changes). Can you do a better job formalizing what your alternative is?
Except she doesn’t. She’ll give the same answer on Monday as she will on Tuesday, because she doesn’t learn anything by waking up.
Yes, this is very alarming, considering this is a forum for aspiring rationalists.
I disagree; but I’ve already given my reasons.
Which of your down-voted statements were correct?
Well, I got −6 for this statement: “P(monday and heads)=1/2. P(monday and tails)=1/4. P(tuesday and tails)=1/4. Remember, these have to add to 1.”
Initially there is a 50% chance for heads and 50% chance for tails. Given heads, it’s monday with certainty. So, P(heads)=1/2, p(monday | heads)=1.
Do you dispute either of those?
Similarly, p(tails)=1/2, p(monday | tails)=1/2. p(tuesday | tails)=1/2.
Do you dispute either of those?
The above are all of the probabilities you need to know. From them, you can derive anything that is of interest here.
For example, on an awakening p(monday)=p(monday|tails)p(tails) + p(monday|heads) p(heads)=1/4+1/2=3/4
p(monday and heads)=p(heads)*p(monday|heads)=1/2
etc.
Re: “P(monday and heads)=1/2. P(monday and tails)=1/4. P(tuesday and tails)=1/4. Remember, these have to add to 1.”
Yes, but those Ps are wrong—they should all be 1⁄3.
My assumptions and use of probability laws are clearly stated above. Tell me where I made a mistake, otherwise just saying “you’re wrong” is not going to move things forward.
Well, the correct sum is this one:
“Suppose this experiment were repeated 1,000 times. We would expect to get 500 heads and 500 tails. So Beauty would be awoken 500 times after heads on Monday, 500 times after tails on Monday, and 500 times after tails on Tuesday. In other words, only in a third of the cases would heads precede her awakening. So the right answer for her to give is 1⁄3. This is the correct answer from Beauty’s perspective.”
That gives:
P(monday and heads)=500/1500. P(monday and tails)=500/1500. P(tuesday and tails)=500/1500.
You appear to have gone wrong by giving a different answer—based on a misinterpretation of the meaning of the interview question, it appears.
So you are not willing to tell me where I made a mistake?
P(heads)=1/2, p(monday | heads)=1. Which one of these is wrong?
You’re using expected frequencies to estimate a probability, apparently. But you’re counting the wrong thing. What you are calling P(monday and heads) is not that. There is a problem with your denominator. Think about it. Your numerator has a maximum value of 1000 (if the experiment was repeated 1000 times). Your denominator has a maximum value of 2000. If the maximum possible values of the numerator and denominator do not match, there is a problem. You have an outcome-dependent denominator. Try taking expectation of that. You won’t get what you think you’ll get.
Re: “If the maximum possible values of the numerator and denominator do not match, there is a problem.
The total possible number of awakenings is 2000.
That represents all tails—e.g.:
P(monday and heads) = 0/2000; P(monday and tails) = 1000/2000; P(tuesday and tails) = 1000/2000;
These values add up to 1 - i.e. the total numerators add up to the commonn denominator. That is the actual constraint. The maximum possible value of the numerator in each individual fraction is permitted to be smaller than the common denominator—that is not indicative of a problem.
Oh, it is a huge problem. It proves that your ratio isn’t of the form # of events divided by # of trials. Your ratio is something else. The burden is on you to prove that it actually converges to a probability as the number of trials goes to infinity.
Using cell counts and taking a ratio leads to a probability as the number of trials goes to infinity if you have independent draws. You don’t. You have a strange dependence in there that messes things up. Standard theory doesn’t hold. Your thing there is estimating something, you just don’t know what it is
The total number of events (statements by Beauty) adds up to the total number of trials (interviews).
You should not expect the number of statements by beauty on Monday to add up to the total number of interviews alltogether. It adds up to the number of interviews on Monday. This is not very complicated.
Do you have to make a condescending remark every time you respond? You told me things that I already know, and then said “This is not very complicated.” Great, but nothing accomplished.
You are using an estimator that is valid when you have counts from independent trials. Coin flips are independent here, but interviews are not. You need to take that into account.
It is the plain truth. I don’t know why you are asking such silly questions in public. Maybe you have a weak background in this sort of maths. Or maybe you just don’t like admitting that you posted a whole bunch of inaccurate nonsense—and so keep digging yourself deeper in.
You show no sign of being able to understand your problems—so it seems to me as though there is little point in continuing to point them out. You can’t say I didn’t try to help you sort yourself out.
Well, I have a phd in biostatistics and teach Bayesian data analysis at the University of Pennsylvania, so I either have background in such matters or Penn isn’t real careful on who they hire.
The fact that I am very careful about these kinds of problems is what lead me to discover the flaw in the 1⁄3 argument—it wasn’t obvious to me at first.
If true, a good job you haven’t supplied your real name, then—or your friends and colleagues might come across this thread.
Do you find that people generally think more clearly after they’ve been insulted?
Do you find that you think more clearly after you’ve been insulted?
Hi, Nancy! I haven’t researched this issue. I imagine the results would depend on the details of the situation, the relative status of the participants, etc. I recommend you consult a social psychologist—if you are sincerely looking for answers.
NancyLebovitz was being oblique. I believe her point was that your remarks were not useful for the purpose of improving neq1′s state of knowledge.
I would add that they were also not useful for the purpose of entertaining the lurkers, if the downvotes are anything to go by.
Uh—improving neq1′s state of knowledge was not the intended purpose of that post.
I have already written literally dozens of posts attempting to improving the state of knowledge of other participants on this thread. That post was publicly explaining why I am now likely to stop—just so there is no subsequent confusion about the issue.
I think
would have gone over better.
Right—but I call a spade a spade, don’t beat about the bush, say what I think—etc.
Insulating others from what I think in order to protect their egos is not my style. If I did that people would always be wondering if I meant what I said—or whether I was shielding them from my true opinions in order to protect their egos. In the long run, it is best to just speak the truth, as I see it, IMO. At least then, others know where I stand.
There are a lot of approaches one can take when interacting with other people. Your approach leads me to not want to make your acquaintance. The same isn’t true for most of the other people here, even the ones who disagree with me.
In that case, I would recommend you shut up as soon as possible. If, as you said,
then stop wasting your time.
Thanks for your proposal about how to optimise my time management.
I think you are best leaving that issue to me, though—I have more relevant information about the topic than you do.
In the spirit of experimentation, I’m going to try giving up being oblique.
My primary motivation is not to do you a favor. My purpose is also not to protect neq1.
It is to convey that the purpose of this website is to work on thinking clearly, to some extent to further the creation of FAI, and also to improve skills at living. There’s also a little pleasant socializing.
However, insulting people (or having an atmosphere where insults are accepted) does not further any of the purposes of the site.
I believe that people generally think less well when they’ve been insulted. Also, I’ve been online for a long time. Insults are pretty much similar to each other—in other words, they’re noise, not signal so far as anything about the world generally is concerned.. They’re signal about emotional state and/or attempted dominance, but (as should be clear from the conversation so far), not a terribly clear signal.
What’s worse, insults are likely to lead to more insults.
I’m not a moderator, but I’m asking you not to dump hostility here.
Nancy, your beliefs about the average effect of insults on people do not seem to me to be a good reason to avoid bluntly telling people when they are behaving badly. IMO, you are not properly considering the positive effects of pointing out such bad behaviour. If someone behaves badly, and you don’t tell them, they don’t learn. Others might think their behaviour is acceptable. Still others might think you approve of their behaviour—and so on. It is not as though I had not tried all manner of rational argument first. Yes, people might be insulted or offended by someone else pointing out what is going on—if it reflects badly on them, but that is—ultimately—their business.
I’ve told you rather bluntly that I don’t approve of your behavior, though I think I’ve managed to avoid insulting your intelligence or character.
Do you think the world is a better place as a result?
Not especially—and certainly not from my point of view. Alas, I found responding to your comments to be a waste of my time and energy. Especially so with your “oblique” comments. So, overall, I would rather you had not bothered commenting in the first place.
I apologize.
In the spirit of experimentation, I’m going to try giving up being oblique.
My primary motivation is not to do you a favor. My purpose is also not to protect neq1.
It is to convey that the purpose of this website is to work on thinking clearly, to some extent to further the creation of FAI, and also to improve skills at living. There’s also a little pleasant socializing.
However, insulting people (or having an atmosphere where insults are accepted) does not further any of the purposes of the site.
I believe that people generally think less well when they’ve been insulted. Also, I’ve been online for a long time. Insults are pretty much similar to each other—in other words, they’re noise, not signal so far as anything about the world generally is concerned.. They’re signal about emotional state and/or attempted dominance, but (as should be clear from the conversation so far), not a terribly clear signal.
What’s worse, insults are likely to lead to more insults.
I’m not a moderator, but I’m asking you not to dump hostility here.
In the spirit of experimentation, I’m going to try giving up being oblique.
My primary motivation is not to do you a favor. My purpose is also not to protect neq1.
It is to convey that the purpose of this website is to work on thinking clearly, to some extent to further the creation of FAI, and also to improve skills at living. There’s also a little pleasant socializing.
However, insulting people (or having an atmosphere where insults are accepted) does not further any of the purposes of the site.
I believe that people generally think less well when they’ve been insulted. Also, I’ve been online for a long time. Insults are pretty much similar to each other—in other words, they’re noise, not signal so far as anything about the world generally is concerned.. They’re signal about emotional state and/or attempted dominance, but (as should be clear from the conversation so far), not a terribly clear signal.
What’s worse, insults are likely to lead to more insults.
I’m not a moderator, but I’m asking you not to dump hostility here.
In the spirit of experimentation, I’m going to try giving up being oblique.
My primary motivation is not to do you a favor. My purpose is also not to protect neq1.
It is to convey that the purpose of this website is to work on thinking clearly, to some extent to further the creation of FAI, and also to improve skills at living. There’s also a little pleasant socializing.
However, insulting people (or having an atmosphere where insults are accepted) does not further any of the purposes of the site.
I believe that people generally think less well when they’ve been insulted. Also, I’ve been online for a long time. Insults are pretty much similar to each other—in other words, they’re noise, not signal so far as anything about the world generally is concerned.. They’re signal about emotional state and/or attempted dominance, but (as should be clear from the conversation so far), not a terribly clear signal.
What’s worse, insults are likely to lead to more insults.
I’m not a moderator, but I’m asking you not to dump hostility here.
But there are probably an infinite number of propositions that you actually believe, and even an infinite number of relevant propositions that you actually believe. You choose which things that you think to actually say (I’m just assuming that everything you think ‘out loud’ doesn’t get posted to Less Wrong, since I assume you have more thoughts than I’ve observed comments from you). As long as you’re leaving out an infinite amount of information, you might as well also leave out insulting language.
I said he was asking “silly questions”. However, that is true—and was not “insulting language”. If you think I was using “insulting language”, you will have to be more specific about what you mean.
As to the possibility of you making a more general point, IMO, systematically not speaking truths that might cause offense would have bad results—especially for truth-seekers:
“Unfortunately some people take offense more easily than others. Also, some people are offended by true statements.”
http://timtyler.org/political_correctness/
It is the same with pressuring other people to not speak truths that might cause offense. That too, would have—and has had—seriously unpleasant long-term effects.
You didn’t mention that you were likely to stop posting in the thread.
That was implied information: “it seems to me as though there is little point in continuing to point them out. You can’t say I didn’t try to help you sort yourself out.”
Your continued involvement in the conversation was stronger information.
“Likely to stop” is a probabalistic statement. I am still likely to stop posting on this thread soon. I have done my bit to promote the correct answer to this problem. A top level post explains the correct answer in some detail. I feel as though my work here is done.
Were someone else exhibiting similar posting behavior, would you draw the same conclusion? You may sincerely desire to terminate your conversation with neq1, but you appear to cesire* to continue it.
* A “cesire” is a motivator to action that works like a desire even when accompanied by a conflicting desire—much like an alief can induce emotional reactions in the same way as beliefs even in the presence of a contrary belief.
Your question seems vague: Similar posting behavior to what posting behaviour? - and by whom? Would I draw the same conclusion—as which conclusion?
The conclusion that you feel your work is done. Such a state removes the desire to continue responding to neq1, and—as such a desire is the only apparent reason to respond to neq1 - leads to a cessation of posts in the associated thread(s). This has not occurred.
I haven’t argued about the topic of this post for a little while now—and certainly not since writing “I feel as though my work here is done”.
Rather I am here defending my reputation against assaults from people who don’t like my posting style—and seem keen to let everyone else know of their disapproval. I’ll probably give up with that too, soon enough.
Were someone else exhibiting similar posting behavior, would you draw the same conclusion? You may sincerely desire to terminate your conversation with neq1, but you appear to cesire* to continue it.
* A “cesire” is a motivator to action that works like a desire even when accompanied by a conflicting desire—much like an alief can induce emotional reactions in the same way as beliefs even in the presence of a contrary belief.
Actually, I have supplied my real name (in a previous post I linked to my blog, which has my name). I’m confident my colleagues would be in agreement with me.
Or they all should be 1⁄2.
Impossible—if they are to add up to 1.
For Jack’s bookie, I agree, you have to use 1⁄3 – but if you want to calculate a distribution on how much cash Beauty has after the experiment given different betting behavior, it no longer works to treat Monday and Tuesday as mutually exclusive.