7 ms·

The Mathematics of Changing Your Mind

102 points by jonburs 15y ago

kqr2 15y ago
An intuitive expanation of Bayes' theorem: http://yudkowsky.net/rational/bayes http://yudkowsky.net/rational/bayes
- gosub 15y ago
  An Intuitive Explanation of Eliezer Yudkowsky’s Intuitive Explanation of Bayes’ Theorem: http://commonsenseatheism.com/?p=13156 http://commonsenseatheism.com/?p=13156
IvoDankolov 15y ago
I had to do a double take when I saw Bayes's theorem on that illustration - "wait a second, did I read that domain name correctly"? Even more surprising is that they would use the terminology of prior and posterior probabilities and the like, applied correctly. It gives one the impression that the author knows what he is talking about. I don't know how motivating this article would be to someone without training in probability theory, though. I mean, it's all well and good to be factually correct, but showering people outside of the field with technical terms and "weird" claims can, I think, be very counterproductive. Concerning the theorem itself, no effort was actually made to at least rephrase it in a more friendly way. It's not enough to just poke fun at yourself with "(trumpets sound here)". Many people, even with formal mathematical training do not intuitively understand why it is true, and without that, applying it for probability calculations would feel like hollow and mindless mechanical work. To that effect, I also recommend Eliezer's explanation, found at [0]. I must admit that I kind of glossed over it, because I had the correct intuition regardless of not formally knowing the theorem (no idea how to reproduce that, unfortunately, because it would be a pretty good tool), but even so, I found it both funny and informative. And as a bonus, it doesn't assume that you have any rigorous training in mathematics (a common misconception among mathematical explanations). [0] : http://yudkowsky.net/rational/bayes http://yudkowsky.net/rational/bayes
- wnoise 15y ago
  > Even more surprising is that they would use the terminology of prior and posterior probabilities and the like, applied correctly. It gives one the impression that the author knows what he is talking about. Well, the author of the review is John Allen Paulos, a mathematics professor.
- a3_nm 15y ago
  Understanding why Bayes' theorem is true isn't the hardest part. P(A and B) is both P(A) P(B|A) and P(B) P(A|B), and the result follows. I find this derivation easier to remember than the usual formula, by the way.
kilian 15y ago
Is there any way for us paywalled people to read this?
- _delirium 15y ago
  Apart from the obvious (buy a subscription), you can open the link in an incognito/private-browsing session, since the NYT monthly page limit seems to be based on cookies.
- vlisivka 15y ago
  Zap cookies bookmarklet: javascript:(function(){C=document.cookie.split(%22;%20%22);for(d=%22.%22+location.host;d;d=(%22%22+d).substr(1).match(/\..*$/))for(sl=0;sl<2;++sl)for(p=%22/%22+location.pathname;p;p=p.substring(0,p.lastIndexOf('/')))for(i%20in%20C)if(c=C[i]){document.cookie=c+%22;%20domain=%22+d.slice(sl)+%22;%20path=%22+p.slice(1)+%22/%22+%22;%20expires=%22+new%20Date((new%20Date).getTime()-1e11).toGMTString()}})()
- OpenAmazing 15y ago
  Don't they let any user from Google past the paywall? Just google the article title.
- gwern 15y ago
  Apparently, appending ? and then anything to the link usually gets you in. (I like to append "?reddit". It's often true, even.)
tmoertel 15y ago
While I can understand why probability textbooks present Bayes' Theorem as P(A|B) = [ P(B|A) * P(A) ] / P(B), I've always thought the following rewrite makes more sense when the probabilities represent degrees of belief: P(A|B) = P(A) * [ P(B|A) / P(B) ] In other words: (posterior belief) = (prior belief) * (evidence adjustment) Also, while Bayes' Theorem follows from probability axioms, Bayes' Rule for updating beliefs in light of new evidence (which has the same formulation) is justified by Cox's Theorem: http://en.wikipedia.org/wiki/Coxs_theorem http://en.wikipedia.org/wiki/Coxs_theorem

102 points by jonburs 15y ago

kqr2 15y ago

An intuitive expanation of Bayes' theorem: http://yudkowsky.net/rational/bayes http://yudkowsky.net/rational/bayes

gosub 15y ago
An Intuitive Explanation of Eliezer Yudkowsky’s Intuitive Explanation of Bayes’ Theorem: http://commonsenseatheism.com/?p=13156 http://commonsenseatheism.com/?p=13156

IvoDankolov 15y ago

I had to do a double take when I saw Bayes's theorem on that illustration - "wait a second, did I read that domain name correctly"? Even more surprising is that they would use the terminology of prior and posterior probabilities and the like, applied correctly. It gives one the impression that the author knows what he is talking about. I don't know how motivating this article would be to someone without training in probability theory, though. I mean, it's all well and good to be factually correct, but showering people outside of the field with technical terms and "weird" claims can, I think, be very counterproductive. Concerning the theorem itself, no effort was actually made to at least rephrase it in a more friendly way. It's not enough to just poke fun at yourself with "(trumpets sound here)". Many people, even with formal mathematical training do not intuitively understand why it is true, and without that, applying it for probability calculations would feel like hollow and mindless mechanical work. To that effect, I also recommend Eliezer's explanation, found at [0]. I must admit that I kind of glossed over it, because I had the correct intuition regardless of not formally knowing the theorem (no idea how to reproduce that, unfortunately, because it would be a pretty good tool), but even so, I found it both funny and informative. And as a bonus, it doesn't assume that you have any rigorous training in mathematics (a common misconception among mathematical explanations). [0] : http://yudkowsky.net/rational/bayes http://yudkowsky.net/rational/bayes

wnoise 15y ago
> Even more surprising is that they would use the terminology of prior and posterior probabilities and the like, applied correctly. It gives one the impression that the author knows what he is talking about. Well, the author of the review is John Allen Paulos, a mathematics professor.
a3_nm 15y ago
Understanding why Bayes' theorem is true isn't the hardest part. P(A and B) is both P(A) P(B|A) and P(B) P(A|B), and the result follows. I find this derivation easier to remember than the usual formula, by the way.

kilian 15y ago

Is there any way for us paywalled people to read this?

_delirium 15y ago
Apart from the obvious (buy a subscription), you can open the link in an incognito/private-browsing session, since the NYT monthly page limit seems to be based on cookies.
vlisivka 15y ago
Zap cookies bookmarklet: javascript:(function(){C=document.cookie.split(%22;%20%22);for(d=%22.%22+location.host;d;d=(%22%22+d).substr(1).match(/\..*$/))for(sl=0;sl<2;++sl)for(p=%22/%22+location.pathname;p;p=p.substring(0,p.lastIndexOf('/')))for(i%20in%20C)if(c=C[i]){document.cookie=c+%22;%20domain=%22+d.slice(sl)+%22;%20path=%22+p.slice(1)+%22/%22+%22;%20expires=%22+new%20Date((new%20Date).getTime()-1e11).toGMTString()}})()
OpenAmazing 15y ago
Don't they let any user from Google past the paywall? Just google the article title.
gwern 15y ago
Apparently, appending ? and then anything to the link usually gets you in. (I like to append "?reddit". It's often true, even.)

tmoertel 15y ago

While I can understand why probability textbooks present Bayes' Theorem as P(A|B) = [ P(B|A) * P(A) ] / P(B), I've always thought the following rewrite makes more sense when the probabilities represent degrees of belief: P(A|B) = P(A) * [ P(B|A) / P(B) ] In other words: (posterior belief) = (prior belief) * (evidence adjustment) Also, while Bayes' Theorem follows from probability axioms, Bayes' Rule for updating beliefs in light of new evidence (which has the same formulation) is justified by Cox's Theorem: http://en.wikipedia.org/wiki/Coxs_theorem http://en.wikipedia.org/wiki/Coxs_theorem