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)