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PhilGoetz comments on What is Bayesianism? - Less Wrong
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Evidence, as measured in log odds, has the nice property that evidence from independent sources can be combined by adding. Is there any way at all to combine p-values from independent sources? As I understand them, p-values are used to make a single binary decision to declare a theory supported or not, not to track cumulative strength of belief in a theory. They are not a measure of evidence.
Log odds of independent events do not add up, just as the odds of independent events do not multiply. The odds of flipping heads is 1:1, the odds of flipping heads twice is not 1:1 (you have to multiply odds by likelihood ratios, not odds by odds, and likewise you don't add log odds and log odds, but log odds and log likelihood-ratios). So calling log odds themselves "evidence" doesn't fit the way people use the word "evidence" as something that "adds up".
This terminology may have originated here:
http://causalityrelay.wordpress.com/2008/06/23/odds-and-intuitive-bayes/
I'm voting your comment up, because I think it's a great example of how terminology should be chosen and used carefully. If you decide to edit it, I think it would be most helpful if you left your original words as a warning to others :)
Can you be clearer? Log likelihood ratios do add up, so long as the independence criterion is satisfied (ie so long as P(E_2|H_x) = P(E_2|E_1,H_x) for each H_x).
Sure, just edited in the clarification: "you have to multiply odds by likelihood ratios, not odds by odds, and likewise you don't add log odds and log odds, but log odds and log likelihood-ratios".
As long as there are only two H_x, mind you. They no longer add up when you have three hypotheses or more.
Indeed - though I find it very hard to hang on to my intuitive grasp of this!
Here is the post on information theory I said I would write:
http://lesswrong.com/lw/1y9/information_theory_and_the_symmetry_of_updating/
It explains "mutual information", i.e. "informational evidence", which can be added up over as many independent events as you like. Hopefully this will have restorative effects for your intuition!
Don't worry, I have an information theory post coming up that will fix all of this :)