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?
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.