Entertainingly, I feel justified in ignoring your argument and most of the others for the same reason you feel justified in ignoring other arguments.
I got into a discussion about the SB problem a month ago after Mallah mentioned it as related to the red door/blue doors problem. After a while I realized I could get either of 1⁄2 or 1⁄3 as an answer, despite my original intuition saying 1⁄2.
I confirmed both 1⁄2 and 1⁄3 were defensible by writing a computer program to count relative frequencies two different ways. Once I did that, I decided not to take seriously any claims that the answer had to be one or the other, since how could a simple argument overrule the result of both my simple arithmetic and a computer simulation?
A higher level of understanding of an initially mysterious question should translate into knowing why people may disagree, and still insist on answers that you yourself have discarded. You explain away their disagreement as an inferential distance.
Neither of the answers you have arrived at is correct, from my perspective, and I can explain why. So I feel justified in ignoring your argument for ignoring my argument. :)
That a simulation program should compute 1⁄2 for “how many times on average the coin comes up heads per time it is flipped” is simply P(x) in my formalization. It’s a correct but entirely uninteresting answer to something other than the problem’s question.
That your program should compute 1⁄3 for “how many times on average the coin comes up heads per time Beauty is awoken” is also a correct answer to a slightly more subtly mistaken question. If you look at the “Halfer variant” page of my spreadsheet, you will see a probability distribution that also correspond to the same “facts” that yield the 1⁄3 answer, and yet applying the laws of probability to that distribution give Beauty a credence of 1⁄2. The question your program computes an answer to is not the question “what is the marginal probability of x=Heads, conditioning on z=Woken”.
Whereas, from the tables representing the joint probability distribution, I think I now ought to be able to write a program which can recover either answer: the Thirder answer by inputting the “right” model or the Halfer answer by inputting the “wrong” model. In the Halfer model, we basically have to fail to sample on Heads/Tuesday. Commenting out one code line might be enough.
ETA: maybe not as simple as that, now that I have a first cut of the program written; we’d need to count awakenings on monday twice, which makes no sense at all. It does look as if our programs are in fact computing the same thing to get 1⁄3.
Which specific formulation of the Sleeping Beauty problem did you use to work things out? Maybe we’re referring to descriptions of the problem that use different wording; I’ve yet to read a description that’s convinced me that 1⁄2 is an answer to the wrong question. For example, here’s the wiki’s description asks
Beauty wakes up in the experiment and is asked, “With what subjective probability do believe that the coin landed tails?”
Personally, I believe that using the word ‘subjective’ doesn’t add anything here (it just sounds like a cue to think Bayesian-ishly to me, which doesn’t change the actual answer). So I read the question as asking for the probability of the coin landing tails given the experiment’s setup. As it’s asking for a probabiliy, I see it as wholly legitimate to answer it along the lines of ‘how many times on average the coin comes up heads per X,’ where X is one of the two choices you mentioned.
If you ignore the specification that it is Beauty’s subjective probability under discussion, the problem becomes ill-defined—and multiple answers become defensible—depending on whose perspective we take.
The word ‘subjective’ before the word ‘probability’ is empty verbiage to me, so (as I see it) it doesn’t matter whether you or I have subjectivity in mind. The problem’s ill-defined either way; ‘the specification that it is Beauty’s subjective probability’ makes no difference to me.
“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. Yet to the experimenter the correct probability is 1⁄2.”
I think it’s not the change in perspective or subjective identity making a difference, but instead it’s a change in precisely which probability is being asked about. The Wikipedia page unhelpfully conflates the two changes.
It says that the experimenter must see a probability of 1⁄2 and Beauty must see a probability of 1⁄3, but that just ain’t so; there is nothing stopping Beauty from caring about the proportion of coin flips that turn out to be heads (which is 1⁄2), and there is nothing stopping the experimenter from caring about the proportion of wakings for which the coin is heads (which is 1⁄3). You can change which probability you care about without changing your subjective identity and vice versa.
Let’s say I’m Sleeping Beauty. I would interpret the question as being about my estimate of a probability (‘credence’) associated with a coin-flipping process. Having interpreted the question as being about that process, I would answer 1⁄2 - who I am would have nothing to do with the question’s correct answer, since who I am has no effect on the simple process of flipping a fair coin and I am given no new information after the coin flip about the coin’s state.
“What is your credence now for the proposition that our coin landed heads?”
That’s fairly clearly the PROBABILITY NOW of the coin having landed heads—and not the PROPORTION that turn out AT SOME POINT IN THE FUTURE to have landed heads.
Perspective can make a difference—because different observers have different levels of knowledge about the situation. In this case, Beauty doesn’t know whether it is Tuesday or not—but she does know that if she is being asked on Tuesday, then the coin came down tails—and p(heads) is about 0.
In the original problem post, Beauty is asked a specific question, though
It’s not specific enough. It only asks for Beauty’s credence of a coin landing heads—it doesn’t tell her to choose between the credence of a coin landing heads given that it is flipped and the credence of a coin landing heads given a single waking. The fact that it’s Beauty being asked does not, in and of itself, mean the question must be asking the latter probability. It is wholly reasonable for Beauty to interpret the question as being about a coin-flipping process for which the associated probability is 1⁄2.
That’s fairly clearly the PROBABILITY NOW of the coin having landed heads—and not the PROPORTION that turn out AT SOME POINT IN THE FUTURE to have landed heads.
The addition of the word ‘now’ doesn’t magically ban you from considering a probability as a limiting relative frequency.
Perspective can make a difference—because different observers have different levels of knowledge about the situation. In this case, Beauty doesn’t know whether it is Tuesday or not
Agree.
- but she does know that if she is being asked on Tuesday, then the coin came down tails—and p(heads) is about 0.
It’s not clear to me how this conditional can be informative from Beauty’s perspective, as she doesn’t know whether it’s Tuesday or not. The only new knowledge she gets is that she’s woken up; but she has an equal probability (i.e. 1) of getting evidence of waking up if the coin’s heads or if the coin’s tails. So Beauty has no more knowledge than she did on Sunday.
She has LESS knowledge than she had on Sunday in one critical area—because now she doesn’t know what day of the week it is. She may not have learned much—but she has definitely forgotten something—and forgetting things changes your estimates of their liklihood just as much as learning about them does.
She has LESS knowledge than she had on Sunday in one critical area—because now she doesn’t know what day of the week it is. She may not have learned much—but she has definitely forgotten something -
That’s true.
and forgetting things changes your estimates of their liklihood just as much as learning about them does.
I’m not as sure about this. It’s not clear to me how it changes the likelihoods if I sketch Beauty’s situation at time 1 and time 2 as
A coin will be flipped and I will be woken up on Monday, and perhaps Tuesday. It is Sunday.
I have been woken up, so a coin has been flipped. It is Monday or Tuesday but I do not know which.
as opposed to just
A coin will be flipped and I will be woken up on Monday, and perhaps Tuesday.
I have been woken up, so a coin has been flipped. It is Monday or Tuesday but I do not know which.
(Edit to clarify—the 2nd pair of statements is meant to represent roughly how I was thinking about the setup when writing my earlier comment. That is, it’s evident that I didn’t account for Beauty forgetting what day of the week it is in the way timtyler expected, but at the same time I don’t believe that made any material difference.)
I read it as “What is your credence”, which is supposed to be synonymous with “subjective probability”, which—and this is significant—I take to entail that Beauty must condition on having been woken (because she conditions on every piece of information known to her).
In other words, I take the question to be precisely “What is the probability you assign to the coin having come up heads, taking into account your uncertainty as to what day it is.”
Ahhhh, I think I understand a bit better now. Am I right in thinking that your objection is not that you disapprove of relative frequency arguments in themselves, but that you believe the wrong relative frequency/frequencies is/are being used?
Right up until your reply prompted me to write a program to check your argument, I wasn’t thinking in terms of relative frequencies at all, but in terms of probability distributions.
I haven’t learned the rules for relative frequencies yet (by which I mean thing like “(don’t) include counts of variables that have a correlation of 1 in your denominator”), so I really have no idea.
Here is my program—which by the way agrees with neq1′s comment here, insofar as the “magic trick” which will recover 1⁄2 as the answer consists of commenting out the TTW line.
However, this seems perfectly nonsensical when transposed to my spreadsheet: zeroing out the TTW cell at all means I end up with a total probability mass less than 1. So, I can’t accept at the moment that neq1′s suggestion accords with the laws of probability—I’d need to learn what changes to make to my table and why I should make them.
from random import shuffle, randint
flips=1000
HEADS=0
TAILS=1
# individual cells
HMW = HTW = HMD = HTD = 0.0
TMW = TTW = TMD = TTD = 0.0
def run_experiment():
global HMW, HTW, HMD, HTD, TMW, TTW, TMD, TTD
coin = randint(HEADS,TAILS)
if (coin == HEADS):
# wake Beauty on monday
HMW+=1
# drug Beauty on Tuesday
HTD+=1
if (coin == TAILS):
# wake Beauty on monday
TMW+=1
# wake Beauty on Tuesday too
TTW+=1
for i in range(flips):
run_experiment()
print "Total samples where heads divided by total samples ~P(H):",(HMW+HTW+HMD+HTD)/(HMW+HTW+HMD+HTD+TMW+TTW+TMD+TTD)
print "Total samples where woken F(W):",HMW+HTW+TMW+TTW
print "Total samples where woken and heads F(W&H):", HMW+HTW
print "P(W&H)=P(W)P(H|W), so P(H|W)=lim F(W&H)/F(W)"
print "Total samples where woken and heads divided by sample where woken F(H|W):", (HMW+HTW)/(HMW+HTW+TMW+TTW)
Replying again since I’ve now looked at the spreadsheet.
Using my intuition (which says the answer is 1⁄2), I would expect P(Heads, Tuesday, Not woken) + P(Tails, Tuesday, Not woken) > 0, since I know it’s possible for Beauty to not be woken on Tuesday. But the ‘halfer “variant”’ sheet says P(H, T, N) + P(T, T, N) = 0 + 0 = 0, so that sheet’s way of getting 1⁄2 must differ from how my intuition works.
(ETA—Unless I’m misunderstanding the spreadsheet, which is always possible.)
Your program looks good here, your code looks a lot like mine, and I ran it and got ~1/2 for P(H) and ~1/3 for F(H|W). I’ll try and compare to your spreadsheet.
Even in the limit not all relative frequencies are probabilities. In fact, I’m quite sure that in the limit ntails/wakings is not a probability. That’s because you don’t have independent samples of wakings.
Basically, the 2 wakings on tails should be thought of as one waking. You’re just counting the same thing twice. When you include counts of variables that have a correlation of 1 in your denominator, it’s not clear what you are getting back. The thirders are using a relative frequency that doesn’t converge to a probability
Basically, the 2 wakings on tails should be thought of as one waking. You’re just counting the same thing twice.
This is true if we want the ratio of tails to wakings. However...
When you include counts of variables that have a correlation of 1 in your denominator, it’s not clear what you are getting back. The thirders are using a relative frequency that doesn’t converge to a probability
Despite the perfect correlation between some of the variables, one can still get a probability back out—but it won’t be the probability one expects.
Maybe one day I decide I want to know the probability that a randomly selected household on my street has a TV. I print up a bunch of surveys and put them in people’s mailboxes. However, it turns out that because I am very absent-minded (and unlucky), I accidentally put two surveys in the mailboxes of people with a TV, and only one in the mailboxes of people without TVs. My neighbors, because they enjoy filling out surveys so much, dutifully fill out every survey and send them all back to me. Now the proportion of surveys that say ‘yes, I have a TV’ is not the probability I expected (the probability of a household having a TV) - but it is nonetheless a probability, just a different one (the probability of any given survey saying, ‘I have a TV’).
That’s a good example. There is a big difference though (it’s subtle). With sleeping beauty, the question is about her probability at a waking. At a waking, there are no duplicate surveys. The duplicates occur at the end.
That is a difference, but it seems independent from the point I intended the example to make. Namely, that a relative frequency can still represent a probability even if its denominator includes duplicates—it will just be a different probability (hence why one can get 1⁄3 instead of 1⁄2 for SB).
Entertainingly, I feel justified in ignoring your argument and most of the others for the same reason you feel justified in ignoring other arguments.
I got into a discussion about the SB problem a month ago after Mallah mentioned it as related to the red door/blue doors problem. After a while I realized I could get either of 1⁄2 or 1⁄3 as an answer, despite my original intuition saying 1⁄2.
I confirmed both 1⁄2 and 1⁄3 were defensible by writing a computer program to count relative frequencies two different ways. Once I did that, I decided not to take seriously any claims that the answer had to be one or the other, since how could a simple argument overrule the result of both my simple arithmetic and a computer simulation?
I was thinking about that earlier.
A higher level of understanding of an initially mysterious question should translate into knowing why people may disagree, and still insist on answers that you yourself have discarded. You explain away their disagreement as an inferential distance.
Neither of the answers you have arrived at is correct, from my perspective, and I can explain why. So I feel justified in ignoring your argument for ignoring my argument. :)
That a simulation program should compute 1⁄2 for “how many times on average the coin comes up heads per time it is flipped” is simply P(x) in my formalization. It’s a correct but entirely uninteresting answer to something other than the problem’s question.
That your program should compute 1⁄3 for “how many times on average the coin comes up heads per time Beauty is awoken” is also a correct answer to a slightly more subtly mistaken question. If you look at the “Halfer variant” page of my spreadsheet, you will see a probability distribution that also correspond to the same “facts” that yield the 1⁄3 answer, and yet applying the laws of probability to that distribution give Beauty a credence of 1⁄2. The question your program computes an answer to is not the question “what is the marginal probability of x=Heads, conditioning on z=Woken”.
Whereas, from the tables representing the joint probability distribution, I think I now ought to be able to write a program which can recover either answer: the Thirder answer by inputting the “right” model or the Halfer answer by inputting the “wrong” model. In the Halfer model, we basically have to fail to sample on Heads/Tuesday. Commenting out one code line might be enough.
ETA: maybe not as simple as that, now that I have a first cut of the program written; we’d need to count awakenings on monday twice, which makes no sense at all. It does look as if our programs are in fact computing the same thing to get 1⁄3.
Which specific formulation of the Sleeping Beauty problem did you use to work things out? Maybe we’re referring to descriptions of the problem that use different wording; I’ve yet to read a description that’s convinced me that 1⁄2 is an answer to the wrong question. For example, here’s the wiki’s description asks
Personally, I believe that using the word ‘subjective’ doesn’t add anything here (it just sounds like a cue to think Bayesian-ishly to me, which doesn’t change the actual answer). So I read the question as asking for the probability of the coin landing tails given the experiment’s setup. As it’s asking for a probabiliy, I see it as wholly legitimate to answer it along the lines of ‘how many times on average the coin comes up heads per X,’ where X is one of the two choices you mentioned.
If you ignore the specification that it is Beauty’s subjective probability under discussion, the problem becomes ill-defined—and multiple answers become defensible—depending on whose perspective we take.
The word ‘subjective’ before the word ‘probability’ is empty verbiage to me, so (as I see it) it doesn’t matter whether you or I have subjectivity in mind. The problem’s ill-defined either way; ‘the specification that it is Beauty’s subjective probability’ makes no difference to me.
The perspective makes a difference:
“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. Yet to the experimenter the correct probability is 1⁄2.”
http://en.wikipedia.org/wiki/Sleeping_Beauty_problem
I think it’s not the change in perspective or subjective identity making a difference, but instead it’s a change in precisely which probability is being asked about. The Wikipedia page unhelpfully conflates the two changes.
It says that the experimenter must see a probability of 1⁄2 and Beauty must see a probability of 1⁄3, but that just ain’t so; there is nothing stopping Beauty from caring about the proportion of coin flips that turn out to be heads (which is 1⁄2), and there is nothing stopping the experimenter from caring about the proportion of wakings for which the coin is heads (which is 1⁄3). You can change which probability you care about without changing your subjective identity and vice versa.
Let’s say I’m Sleeping Beauty. I would interpret the question as being about my estimate of a probability (‘credence’) associated with a coin-flipping process. Having interpreted the question as being about that process, I would answer 1⁄2 - who I am would have nothing to do with the question’s correct answer, since who I am has no effect on the simple process of flipping a fair coin and I am given no new information after the coin flip about the coin’s state.
In the original problem post, Beauty is asked a specific question, though—namely:
“What is your credence now for the proposition that our coin landed heads?”
That’s fairly clearly the PROBABILITY NOW of the coin having landed heads—and not the PROPORTION that turn out AT SOME POINT IN THE FUTURE to have landed heads.
Perspective can make a difference—because different observers have different levels of knowledge about the situation. In this case, Beauty doesn’t know whether it is Tuesday or not—but she does know that if she is being asked on Tuesday, then the coin came down tails—and p(heads) is about 0.
It’s not specific enough. It only asks for Beauty’s credence of a coin landing heads—it doesn’t tell her to choose between the credence of a coin landing heads given that it is flipped and the credence of a coin landing heads given a single waking. The fact that it’s Beauty being asked does not, in and of itself, mean the question must be asking the latter probability. It is wholly reasonable for Beauty to interpret the question as being about a coin-flipping process for which the associated probability is 1⁄2.
The addition of the word ‘now’ doesn’t magically ban you from considering a probability as a limiting relative frequency.
Agree.
It’s not clear to me how this conditional can be informative from Beauty’s perspective, as she doesn’t know whether it’s Tuesday or not. The only new knowledge she gets is that she’s woken up; but she has an equal probability (i.e. 1) of getting evidence of waking up if the coin’s heads or if the coin’s tails. So Beauty has no more knowledge than she did on Sunday.
She has LESS knowledge than she had on Sunday in one critical area—because now she doesn’t know what day of the week it is. She may not have learned much—but she has definitely forgotten something—and forgetting things changes your estimates of their liklihood just as much as learning about them does.
That’s true.
I’m not as sure about this. It’s not clear to me how it changes the likelihoods if I sketch Beauty’s situation at time 1 and time 2 as
A coin will be flipped and I will be woken up on Monday, and perhaps Tuesday. It is Sunday.
I have been woken up, so a coin has been flipped. It is Monday or Tuesday but I do not know which.
as opposed to just
A coin will be flipped and I will be woken up on Monday, and perhaps Tuesday.
I have been woken up, so a coin has been flipped. It is Monday or Tuesday but I do not know which.
(Edit to clarify—the 2nd pair of statements is meant to represent roughly how I was thinking about the setup when writing my earlier comment. That is, it’s evident that I didn’t account for Beauty forgetting what day of the week it is in the way timtyler expected, but at the same time I don’t believe that made any material difference.)
I read it as “What is your credence”, which is supposed to be synonymous with “subjective probability”, which—and this is significant—I take to entail that Beauty must condition on having been woken (because she conditions on every piece of information known to her).
In other words, I take the question to be precisely “What is the probability you assign to the coin having come up heads, taking into account your uncertainty as to what day it is.”
Ahhhh, I think I understand a bit better now. Am I right in thinking that your objection is not that you disapprove of relative frequency arguments in themselves, but that you believe the wrong relative frequency/frequencies is/are being used?
Right up until your reply prompted me to write a program to check your argument, I wasn’t thinking in terms of relative frequencies at all, but in terms of probability distributions.
I haven’t learned the rules for relative frequencies yet (by which I mean thing like “(don’t) include counts of variables that have a correlation of 1 in your denominator”), so I really have no idea.
Here is my program—which by the way agrees with neq1′s comment here, insofar as the “magic trick” which will recover 1⁄2 as the answer consists of commenting out the TTW line.
However, this seems perfectly nonsensical when transposed to my spreadsheet: zeroing out the TTW cell at all means I end up with a total probability mass less than 1. So, I can’t accept at the moment that neq1′s suggestion accords with the laws of probability—I’d need to learn what changes to make to my table and why I should make them.
Replying again since I’ve now looked at the spreadsheet.
Using my intuition (which says the answer is 1⁄2), I would expect P(Heads, Tuesday, Not woken) + P(Tails, Tuesday, Not woken) > 0, since I know it’s possible for Beauty to not be woken on Tuesday. But the ‘halfer “variant”’ sheet says P(H, T, N) + P(T, T, N) = 0 + 0 = 0, so that sheet’s way of getting 1⁄2 must differ from how my intuition works.
(ETA—Unless I’m misunderstanding the spreadsheet, which is always possible.)
Yeah, that “Halfer variant” was my best attempt at making sense of the 1⁄2 answer, but it’s not very convincing even to me anymore.
That program is simple enough that you can easily compute expectations of your 8 counts analytically.
Your program looks good here, your code looks a lot like mine, and I ran it and got ~1/2 for P(H) and ~1/3 for F(H|W). I’ll try and compare to your spreadsheet.
Well, perhaps because relative frequencies aren’t always probabilities?
Of course. But if I simulate the experiment more and more times, the relative frequencies converge on the probabilities.
Even in the limit not all relative frequencies are probabilities. In fact, I’m quite sure that in the limit ntails/wakings is not a probability. That’s because you don’t have independent samples of wakings.
But if there is a probability to be found (and I think there is) the corresponding relative frequency converges on it almost surely in the limit.
I don’t understand.
I tried to explain it here: http://lesswrong.com/lw/28u/conditioning_on_observers/1zy8
Basically, the 2 wakings on tails should be thought of as one waking. You’re just counting the same thing twice. When you include counts of variables that have a correlation of 1 in your denominator, it’s not clear what you are getting back. The thirders are using a relative frequency that doesn’t converge to a probability
This is true if we want the ratio of tails to wakings. However...
Despite the perfect correlation between some of the variables, one can still get a probability back out—but it won’t be the probability one expects.
Maybe one day I decide I want to know the probability that a randomly selected household on my street has a TV. I print up a bunch of surveys and put them in people’s mailboxes. However, it turns out that because I am very absent-minded (and unlucky), I accidentally put two surveys in the mailboxes of people with a TV, and only one in the mailboxes of people without TVs. My neighbors, because they enjoy filling out surveys so much, dutifully fill out every survey and send them all back to me. Now the proportion of surveys that say ‘yes, I have a TV’ is not the probability I expected (the probability of a household having a TV) - but it is nonetheless a probability, just a different one (the probability of any given survey saying, ‘I have a TV’).
That’s a good example. There is a big difference though (it’s subtle). With sleeping beauty, the question is about her probability at a waking. At a waking, there are no duplicate surveys. The duplicates occur at the end.
That is a difference, but it seems independent from the point I intended the example to make. Namely, that a relative frequency can still represent a probability even if its denominator includes duplicates—it will just be a different probability (hence why one can get 1⁄3 instead of 1⁄2 for SB).
Ok, yes, sometimes relative frequencies with duplicates can be probabilities, I agree.