neq1 comments on Updating, part 1: When can you change your mind? The binary model - Less Wrong

11 Post author: PhilGoetz 13 May 2010 05:55PM

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Comment author: cupholder 14 May 2010 03:14:20PM 3 points [-]

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?

Comment author: neq1 14 May 2010 03:27:17PM 0 points [-]

Well, perhaps because relative frequencies aren't always probabilities?

Comment author: cupholder 14 May 2010 03:49:36PM *  1 point [-]

Of course. But if I simulate the experiment more and more times, the relative frequencies converge on the probabilities.

> 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
Comment author: neq1 14 May 2010 04:24:10PM *  0 points [-]

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.

Comment author: cupholder 14 May 2010 04:38:52PM 0 points [-]

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.

Comment author: neq1 14 May 2010 04:47:27PM 0 points [-]

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

Comment author: cupholder 14 May 2010 05:29:40PM 1 point [-]

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').

Comment author: neq1 14 May 2010 05:34:56PM 1 point [-]

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.

Comment author: cupholder 14 May 2010 06:01:46PM 0 points [-]

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).

Comment author: neq1 14 May 2010 06:05:13PM 1 point [-]

Ok, yes, sometimes relative frequencies with duplicates can be probabilities, I agree.

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