If you just apply Bayes rule, you get 1/2.
Apply it to what terms?
I'm not sure what more I can say without starting to repeat myself, too. All I can say at this point, having formalized my reasoning as both a Python program and an analytical table giving out the full joint distribution, is "Where did I make a mistake?"
Where's the bug in the Python code? How do I change my joint distribution?
I like the version of your halfer variant version of your table. I still need to think about your distributions more though. I'm not sure it makes sense to have a variable 'woken that day' for this problem.
So I think I figured this whole thing out. Are people familiar with the type-token distinction and resulting ambiguities? If I have five copies of the book Catcher in the Rye and you ask me how many books I have there is an ambiguity. I could say one or five. One refers to the type, "Catcher in the Rye is a coming of age novel" is a sentence about the type. Five refers to the number of tokens, "I tossed Catcher in the Rye onto the bookshelf" is a sentence about the token. The distinction is ubiquitous and leads to occasional confusion, enough that the subject is at the top of my Less Wrong to-do list. The type token distinction becomes an issue whenever we introduce identical copies and the distinction dominates my views on personal identity.
In the Sleeping Beauty case, the amnesia means the experience of waking up on Monday and the experience of waking up on Tuesday, while token-distinct are type-identical. If we decide the right thing to update on isn't the token experience but the type experience: well the calculations are really easy. The type experience "waking up" has P=1 for heads and tails. So the prior never changes. I think there are some really good reasons for worrying about types rather than tokens in this context but won't go into until I make sure the above makes sense to someone.
Makes sense to me.
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.
In your example the experimenter has learned whether you have cancer. And she reflects that knowledge in the structure of the experiment: you are woken up 9 times if you have the disease.
Set aside the amnesia effects of the drug for a moment, and consider the experimental setup as a contorted way of imparting the information to the patient. Then you'd agree that with full memory, the patient would have something to update on? As soon as the second day. So there is, normally, an information flow in this setup.
What the amnesia does is selectively impair the patient's ability to condition on available information. it does that in a way which is clearly pathological, and results in the counter-intuitive reply to the question "conditioning on a) your having woken up and b) your inability to tell what day it is, what is your credence"? We have no everyday intuitions about the inferential consequences of amnesia.
Knowing about the amnesia, we can argue that Beauty "shouldn't" condition on being woken up. But if she does, she'll get that strange result. If she does have cancer, she is more likely to be woken up multiple times than once, and being woken up at all does have some evidential weight.
All this, though, being merely verbal aids as I try to wrap my head around the consequences of the math. And therefore to be taken more circumspectly than the math itself.
If she does condition on being woken up, I think she still gets 1/2. I hate to keep repeating arguments, but what she knows when she is woken up is that she has been woken up at least once. If you just apply Bayes rule, you get 1/2.
If conditioning causes her to change her probability, it should do so in such a way that makes her more accurate. But as we see in the cancer problem, people with cancer give the same answer as people without.
Then you'd agree that with full memory, the patient would have something to update on?
Yes, but then we wouldn't be talking about her credence on an awakening. We'd be talking about her credence on first waking and second waking. We'd treat them separately. With amnesia, 2 wakings are the same as 1. It's really just one experience.
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.
Unlike Jack, I'm pessimistic about your proposal. I've already changed my mind not once but twice.
The interesting aspect is that this doesn't feel like I'm vacillating. I have gone from relying on a vague and unreliable intuition in favor of 1/3 qualified with "it depends", to being moderately certain that 1/2 was unambiguously correct, to having worked out how I was allocating all of the probability mass in the original problem and getting back 1/3 as the answer that I cannot help but think is correct. That, plus the meta-observation that no-one, including people I've asked directly (including yourself), has a rebuttal to my construction of the table, is leaving me with a higher degree of confidence than I previously had in 1/3.
It now feels as if I'm justified to ignore pretty much any argument which is "merely" a verbal appeal to one intuition or the other. Either my formalization corresponds to the problem as verbally stated or it doesn't; either my math is correct or it isn't. "Here I stand, I can no other" - at least until someone shows me my mistake.
Morendil,
This is strange. It sounds like you have been making progress towards settling on an answer, after discussion with others. That would suggest to me that discussion can move us towards consensus.
I like your approach a lot. It's the first time I've seen the thirder argument defended with actually probability statements. Personally, I think there shouldn't be any probability mass on 'not woken', but that is something worth thinking about and discussing.
One thing that I think is odd. Thirders know she has nothing to update on when she is woken, because they admit she will give the same answer, regardless of if it's heads or tails. If she really had new information that is correlated with the outcome, her credence would move towards heads when heads, and tails when tails.
Consider my cancer intuition pump example. Everyone starts out thinking there is a 50% chance they have cancer. Once woken, regarldess of if they have cancer or not, they all shift to 90%. Did they really learn anything about their disease state by being woken? If they did, those with cancer would have shifted their credence up a bit, and those without would have shifted down. That's what updating is.
Even in the limit not all relative frequencies are probabilities.
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.
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.
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
Of course. But if I simulate the experiment more and more times, the relative frequencies converge on the probabilities.
= 27d567bc2d48d9f40e9ccc20ea370b15)
> beauty <- function(n) {
+ + # Number of times the coin comes up tails
+ ntails <- 0
+ + # Number of times SB wakes up
+ wakings <- 0
+ + for (i in 1:n) {
+ + # It's Sunday, flip the coin, 0 is heads, 1 is tails
+ coin <- sample(c(0, 1), 1)
+ ntails <- ntails + coin
+ + if (coin == 0) {
+ # Beauty wakes up once, Monday
+ wakings <- wakings + 1
+ } else {
+ # Beauty wakes up Monday, then Tuesday
+ wakings <- wakings + 2
+ }
+ }
+ + return(c(ntails / wakings, ntails / n))
+ + }
> beauty(5)
[1] 0.1666667 0.2000000
> beauty(50)
[1] 0.375 0.600
> beauty(500)
[1] 0.3036212 0.4360000
> beauty(5000)
[1] 0.3314614 0.4958000
> beauty(50000)
[1] 0.3336354 0.5006800
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.
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?
Well, perhaps because relative frequencies aren't always probabilities?
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= b715d02e85f7b3b4ae65ca3d3e450868)
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Huh? If tails, then Beauty is (always) woken on Monday. Why do you have probability=1/2 there?
(likewise for Tuesday)
The probability represents how she should see things when she wakes up.
She knows she's awake. She knows heads had probability 0.5. She knows that, if it landed heads, it's Monday with probability 1. She knows that, if it landed tails, it's either Monday or Tuesday. Since there is no way for her to distinguish between the two, she views them as equally likely. Thus, if tails, it's Monday with probability 0.5 and Tuesday with probability 0.5.