Comments:
Log base ten may be more intuitive for conversion purposes. Then adding another 9 corresponds to adding 1.
"Five times more likely" should overflow for probabilities greater than 0.2. This is because the terminology "times more likely" is usually used in the context of decision-making, so it manipulates the linear probabilities because that's what goes into the expected utility.
Yeah, I was definitely thinking about that. The mathematician in me won out in the end.
It occurs to me that a lot of people have probably thought about this, and they have alternately used base 2, base e, and base 10. Unless we get the entire LW community to standardize on one base, we won't be able to coherently communicate with one another using log-probabilities, and therefore log-probabilities will stay relegated to the dustbin.
base 2 - advantages, we can talk about N bytes' worth of evidences.
base e - mathematician's base
base 10 - common layperson c...
(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)