Comment author:Academian
17 March 2010 01:46:03PM
*
4 points
[-]

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".

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 :)

Comment author:JGWeissman
17 March 2010 04:53:32PM
0 points
[-]

By "evidence", I refer to events that change an agent's strength of belief in a theory, and the measure of evidence is the measure of this change in belief, that is, the likelihood-ratio and log likelihood-ratio you refer to.

I never meant for "evidence" to refer to the posterior strength of belief. "Log odds" was only meant to specify a particular measurement of strength in belief.

Comment author:ciphergoth
17 March 2010 02:44:00PM
*
0 points
[-]

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).

Comment author:Academian
17 March 2010 02:56:52PM
2 points
[-]

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".

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!

## Comments (211)

Best*4 points [-]Log odds of independent events

do notadd 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 :)

By "evidence", I refer to events that change an agent's strength of belief in a theory, and the measure of evidence is the measure of this change in belief, that is, the likelihood-ratio and log likelihood-ratio you refer to.

I never meant for "evidence" to refer to the posterior strength of belief. "Log odds" was only meant to specify a particular measurement of strength in belief.

*0 points [-]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 :)