It seems to me that this doesn't have any real advantage over odds ratios. If I want to do a Bayesian update, I multiply the odds by the relative likelihood. In the example in the article (1/10,000 chance of having the disease, 3% false positive, and 1% false negative), You just take 1:9999 and multiply it by 0.99/0.03 = 33:1 for each successful test. Then you have 33:9999 = 1:303, then 33:303 = 11:101, and finally 363:101 for the final test. Then to change back, you just take 363/(363+101) = 78.23%. The calculations are slower (two multiplications vs. one addition), but it's much easier and more intuitive to convert between them and traditional probabilities.
What you've described is in fact, exactly the same thing as log-odds - they're simply separated by a logarithm/exponentiation. Thus, all the multiplications you describe are the counterpart of the additions I describe. I agree, we could work with odds ratio, without taking the logarithm - but using logarithms has the benefit of linearizing the probability space. The distance between 1 L% and 5 L% is the same as the distance between 10 L% and 14 L%, but you wouldn't know it by looking at 2.72:1 and 150:1 versus 22,000:1 and 1,200,000:1.
(I wrote this post for my own blog, and given the warm reception, I figured it would also be suitable for the LW audience. It contains some nicely formatted equations/tables in LaTeX, hence I've left it as a dropbox download.)
Logarithmic probabilities have appeared previously on LW here, here, and sporadically in the comments. The first is a link to a Eliezer post which covers essentially the same material. I believe this is a better introduction/description/guide to logarithmic probabilities than anything else that's appeared on LW thus far.
Introduction:
Our conventional way of expressing probabilities has always frustrated me. For example, it is very easy to say nonsensical statements like, “110% chance of working”. Or, it is not obvious that the difference between 50% and 50.01% is trivial compared to the difference between 99.98% and 99.99%. It also fails to accommodate the math correctly when we want to say things like, “five times more likely”, because 50% * 5 overflows 100%.
Jacob and I have (re)discovered a mapping from probabilities to log- odds which addresses all of these issues. To boot, it accommodates Bayes’ theorem beautifully. For something so simple and fundamental, it certainly took a great deal of google searching/wikipedia surfing to discover that they are actually called “log-odds”, and that they were “discovered” in 1944, instead of the 1600s. Also, nobody seems to use log-odds, even though they are conceptually powerful. Thus, this primer serves to explain why we need log-odds, what they are, how to use them, and when to use them.
Article is here (Updated 11/30 to use base 10)