I’m saying that there exists a pair of statements (A, B) such that P(A&B|Person says A&B) > P(A|Person says A).
As an example, suppose I told you that Boston once flooded with molasses. This is an unlikely statement, and seems like the sort of thing someone would tell you as a joke.
Now suppose I instead gave you an entire article. That’s pretty far to go for a joke. Each detail in that article is unlikely, but it’s just as unlikely that I’d make up those particular details in addition to being unlikely that I’d be making up that many details in the first place. As such, you’d be more likely to think I’m telling the truth.
Now suppose I hand you a copy of Wikipedia, and point out that article. I might make up a single article. The Onion makes up silly stuff like that all the time. But there’s no way someone will write an entire encyclopedia that generally seems sensible and self-consistent, just so you would believe that one article on a molasses flood. A priori, the idea of Wikipedia being almost all true is absurd, but then, so is the idea that someone would write that exact encyclopedia.
What makes it unbelievable that Boston flooded with molasses is the implied scale. The article is about several blocks flooding, not about “Boston” flooding; the original claim remains unbelievable.
Also, but unrelated, you need to remember that if other statements can increase credibility, they can also reduce it. If you told me Boston was flooded with molasses on the scale implied by that statement, and then directed me to a site that had some good articles but also promoted perpetual motion machines, I wouldn’t give the claim any more credibility. It is true that nobody would make all that stuff up for a joke, but people can make up huge quantities of stuff under self-delusion.
if other statements can increase credibility, they can also reduce it.
Sure, but it’s utterly unsurprising that there exists a B such that P(A&B) P(A) is more surprising, which is why I’d asked for an example of what DanielLC had in mind by it.
Your probability theory here is flawed. The question is not about P(A&B), the probability that both are true, but about P(A|B), the probability that A is true given that B is true. If A is “has cancer” and B is “cancer test is positive”, then we calculate P(A|B) as P(B|A)P(A)/P(B); that is, if there’s a 1/1000 chance of cancer and and the test is right 99⁄100, then P(A|B) is .99.001/(.001.99+.999.01) which is about 1 in 10.
I suppose, given the context, I should say out loud that it wasn’t me, both because I don’t find it downvoteworthy and because I make a practice of not downvoting comments that reply to mine or that I reply to.
I endorse not trying to read much into one or two downvotes… the voting behavior of arbitrarily selected individuals in a group like this doesn’t necessarily mean much.
The way to fix the formatting is to use a \ in front of the asterisk whenever you want to actually display it. This is also necessary for underscores, which some people use in their usernames.
I didn’t downvote it, and don’t have interesting speculation as to why it was downvoted.
Fair enough. I agree that for all B where saying B legitimately increases a speaker’s credibility, observing a speaker saying (A & B) legitimately gives me more confidence in A than the same speaker just saying (A).
I’m saying that there exists a pair of statements (A, B) such that P(A&B|Person says A&B) > P(A|Person says A).
As an example, suppose I told you that Boston once flooded with molasses. This is an unlikely statement, and seems like the sort of thing someone would tell you as a joke.
Now suppose I instead gave you an entire article. That’s pretty far to go for a joke. Each detail in that article is unlikely, but it’s just as unlikely that I’d make up those particular details in addition to being unlikely that I’d be making up that many details in the first place. As such, you’d be more likely to think I’m telling the truth.
Now suppose I hand you a copy of Wikipedia, and point out that article. I might make up a single article. The Onion makes up silly stuff like that all the time. But there’s no way someone will write an entire encyclopedia that generally seems sensible and self-consistent, just so you would believe that one article on a molasses flood. A priori, the idea of Wikipedia being almost all true is absurd, but then, so is the idea that someone would write that exact encyclopedia.
What makes it unbelievable that Boston flooded with molasses is the implied scale. The article is about several blocks flooding, not about “Boston” flooding; the original claim remains unbelievable.
Also, but unrelated, you need to remember that if other statements can increase credibility, they can also reduce it. If you told me Boston was flooded with molasses on the scale implied by that statement, and then directed me to a site that had some good articles but also promoted perpetual motion machines, I wouldn’t give the claim any more credibility. It is true that nobody would make all that stuff up for a joke, but people can make up huge quantities of stuff under self-delusion.
Sure, but it’s utterly unsurprising that there exists a B such that P(A&B) P(A) is more surprising, which is why I’d asked for an example of what DanielLC had in mind by it.
Your probability theory here is flawed. The question is not about P(A&B), the probability that both are true, but about P(A|B), the probability that A is true given that B is true. If A is “has cancer” and B is “cancer test is positive”, then we calculate P(A|B) as P(B|A)P(A)/P(B); that is, if there’s a 1/1000 chance of cancer and and the test is right 99⁄100, then P(A|B) is .99.001/(.001.99+.999.01) which is about 1 in 10.
Can anyone explain why the parent was downvoted? I don’t get it. I hope there’s a better reason than the formatting fail.
I suppose, given the context, I should say out loud that it wasn’t me, both because I don’t find it downvoteworthy and because I make a practice of not downvoting comments that reply to mine or that I reply to.
I endorse not trying to read much into one or two downvotes… the voting behavior of arbitrarily selected individuals in a group like this doesn’t necessarily mean much.
The way to fix the formatting is to use a \ in front of the asterisk whenever you want to actually display it. This is also necessary for underscores, which some people use in their usernames.
I didn’t downvote it, and don’t have interesting speculation as to why it was downvoted.
That’s fair.
Fair enough. I agree that for all B where saying B legitimately increases a speaker’s credibility, observing a speaker saying (A & B) legitimately gives me more confidence in A than the same speaker just saying (A).