cupholder comments on Updating, part 1: When can you change your mind? The binary model

[cupholder]

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

[neq1]

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.

[cupholder]

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.

[neq1]

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

[cupholder]

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’).

[neq1]

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.

[cupholder]

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).

[neq1]

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