The Mathematics of Changing Your Mind(nytimes.com)
nytimes.com
The Mathematics of Changing Your Mind
http://www.nytimes.com/2011/08/07/books/review/the-theory-that-would-not-die-by-sharon-bertsch-mcgrayne-book-review.html?pagewanted=all
4 comments
> 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.
Well, the author of the review is John Allen Paulos, a mathematics professor.
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.
An intuitive expanation of Bayes' theorem:
http://yudkowsky.net/rational/bayes
http://yudkowsky.net/rational/bayes
An Intuitive Explanation of Eliezer Yudkowsky’s Intuitive Explanation of Bayes’ Theorem:
http://commonsenseatheism.com/?p=13156
http://commonsenseatheism.com/?p=13156
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
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
Is there any way for us paywalled people to read this?
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.
Apparently, appending ? and then anything to the link usually gets you in. (I like to append "?reddit". It's often true, even.)
Don't they let any user from Google past the paywall? Just google the article title.
Zap cookies bookmarklet:
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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