In the example with Bob, surely the odds of Bob having a crush on you after winking (2:1) should be higher than a random person winking at you (given as 10:1), as we already have reason to suspect that Bob is more likely to have a crush on you than some random person not part of the six.

Not if you consider that the 1:5 figure constrains that ONLY one person among the six has a crush on you. If you learn for a fact one does, you’ll also immediately know the others all don’t. Which is not true for a random selection of students—you could randomly pick six that all have a crush on you.
Bob belongs to a group in which you know for a fact five people DON’T have a crush on you. So you have evidence lowering Bob’s odds relative to a random winker.

Either that, or it doesn’t matter how many actually have a crush on you, you’re looking for the specific one you have definite evidence about. For thisy, a random winker is not qualified to enter the comparison at all—if they’re not one of the six, they’re not the person you’re looking for. So Bob might have a crush on you AND not be the person you’re looking for, although his odds are higher than those of the other five you don’t have any evidence about.

That’s the interpretations that make the math not wrong, anyway. If you only know that “at least one of them has a crush on me” and more than one could potentially satisfy your search criteria, the 1:5 figure is not the right odds.

“you’ll also immediately know the others all don’t”

No. Receiving an anonymous love note from among the 6 in NO WAY informs you that 5 of the 6 DON’T have a crush on you. All it does is take the unspecified prior (rate of these 6 humans having a crush on you), and INCREASE it for all 6 of them.

@irmckenzie is right. There’s no way you get < 10:1 with MORE positive (confirmatory) evidence for Bob than a random stranger. All positive evidence HAS TO make a rational mind MORE certain the thing is true. Weak evidence, like the letter, which informs that AT LEAST 1 in 6 has a crush, should move a rational mind LESS than strong evidence, like the wink, but it must move it all the same, and in the affirmative direction.

This is obviously correct. The error was that Rob interpreted the evidence incorrectly. Getting an anonymous letter DOES NOT inform a rational mind that Bob has 1:5 odds of crushing. It informs the rational mind that AT LEAST ONE of the 6 classmates has a crush on you. It DOES NOT inform a rational mind that 5 of the 6 classmates DO NOT have a crush on you. I also hated this. Obviously, two pieces of evidence should make Bob MORE LIKELY to have a crush on you that one. There’s no baseline rate of humans having a crush on us, so the real prior isn’t in the problem.

You are right. Seems like there is an error in this example and a main problem is not with a prior 1:5 odds, problem is with a bad phrasing and confusion between “crush when winked” odds and “wink likelihood ratio”.

you get winked at by people ten times as often when they have a crush on you

Is a statement about likelihood ratio (or at least can be interpreted that way) - P(wink|crush):P(wink|!crush)=10:1

And in final calculation likelihood is used and its correct according to Bayes Rule

To change our mind from the 1:5 prior odds in response to the evidence’s 10:1 likelihood ratio, we multiply the left sides together and the right sides together

While a statement

the 10:1 odds in favor of “a random person who winks at me has a crush on me”

is a statement about odds P(crush|wink):P(!crush|wink)=10:1 and to apply a Bayes rule to it as if it is ratio would be a mistake, but i’m guessing it’s just an author’s error in phrasing of this statement.

In the example with Bob, surely the odds of Bob having a crush on you after winking (2:1) should be

higherthan a random person winking at you (given as 10:1), as we already have reason to suspect that Bob is more likely to have a crush on you than some random person not part of the six.Not if you consider that the 1:5 figure constrains that ONLY one person among the six has a crush on you. If you learn for a fact one does, you’ll also immediately know the others all don’t. Which is not true for a random selection of students—you could randomly pick six that all have a crush on you. Bob belongs to a group in which you know for a fact five people DON’T have a crush on you. So you have evidence lowering Bob’s odds relative to a random winker.

Either that, or it doesn’t matter how many actually have a crush on you, you’re looking for the specific one you have definite evidence about. For thisy, a random winker is not qualified to enter the comparison at all—if they’re not one of the six, they’re not the person you’re looking for. So Bob might have a crush on you AND not be the person you’re looking for, although his odds are higher than those of the other five you don’t have any evidence about.

That’s the interpretations that make the math not wrong, anyway. If you only know that “at least one of them has a crush on me” and more than one could potentially satisfy your search criteria, the 1:5 figure is not the right odds.

“you’ll also immediately know the others all don’t”

No. Receiving an anonymous love note from among the 6 in NO WAY informs you that 5 of the 6 DON’T have a crush on you. All it does is take the unspecified prior (rate of these 6 humans having a crush on you), and INCREASE it for all 6 of them.

@irmckenzie is right. There’s no way you get < 10:1 with MORE positive (confirmatory) evidence for Bob than a random stranger. All positive evidence HAS TO make a rational mind MORE certain the thing is true. Weak evidence, like the letter, which informs that AT LEAST 1 in 6 has a crush, should move a rational mind LESS than strong evidence, like the wink, but it must move it all the same, and in the affirmative direction.

This is obviously correct. The error was that Rob interpreted the evidence incorrectly. Getting an anonymous letter DOES NOT inform a rational mind that Bob has 1:5 odds of crushing. It informs the rational mind that AT LEAST ONE of the 6 classmates has a crush on you. It DOES NOT inform a rational mind that 5 of the 6 classmates DO NOT have a crush on you. I also hated this. Obviously, two pieces of evidence should make Bob MORE LIKELY to have a crush on you that one. There’s no baseline rate of humans having a crush on us, so the real prior isn’t in the problem.

What Liliet B said. Low priors will screw with you even after a “definitive” experiment. You might also want to take a look at this: https://www.lesswrong.com/posts/XTXWPQSEgoMkAupKt/an-intuitive-explanation-of-bayes-s-theorem

You are right. Seems like there is an error in this example and a main problem is not with a prior 1:5 odds, problem is with a bad phrasing and confusion between “crush when winked” odds and “wink likelihood ratio”.

Is a statement about likelihood ratio (or at least can be interpreted that way) - P(wink|crush):P(wink|!crush)=10:1

And in final calculation likelihood is used and its correct according to Bayes Rule

While a statement

is a statement about odds P(crush|wink):P(!crush|wink)=10:1 and to apply a Bayes rule to it as if it is ratio would be a mistake, but i’m guessing it’s just an author’s error in phrasing of this statement.