Epistemic status: just a review of a well known math theorem and a brief rant about terminology.
Yesterday I saw another example of this: is just a normalizing constant for the posterior probability, and it's really hard (impossible?) to calculate, so let's switch to log odd probabilities, which are easier and the pesky term is canceled.
Except it's not: , the second term is exactly what you need to get the probability in odd form, and if you have it you can very well calculate the prior for the data.
So please, whatever you write, stop saying that odds are easier. They are possibly more intuitive to manipulate, but they need exactly the same amount of information.
I've always found the notion that "odds are easier" confusing. I'm not sure who they are easier for, but I find reasoning about betting odds confusing and unintuitive. I have a clear feel for what a probability of 0.25 is. I don't have one for what 1:3 means. Maybe most people have greater experience with gambling?
I find if I try using probabilities in Bayes in my head then I make mistakes. If I start at 1/4 probability and get 1 bit of evidence to lower this further then I think “ok, Ill update to 1/8”. If I use odds I start at 1:3, update to 1:6 and get the correct posterior of 1/7.
So essentially I’m constantly going back and forth - like you I find probabilities easier to picture but find odds easier for updates.
If you roll a fair six sided die once, there is a probability of 1/3 of rolling a "1" or a "2". While a probability (#) is followed by a description of what happens, this information is interlaced with the odds:
1:2 means there's 1 set* where you get what you're looking for ("1" or "2") and 2 where you don't ("3" or "4", "5" or "6"). It can also be read as 1/3.
I tried to come up with a specific difference between odds and probability that would suggest where to use one or the other, aside from speed/comfort and multiplication versus addition**, and the only thing I came up with is that you used "0.25" as a probability where I'd have used "1/4".
*This relies on the sets all having equal probability.
**Adding .333 repeating to .25 isn't too hard, .58333 3s repeating. Multiplying those sounds like a mess. (I do not want to multiply anything by .58333 ever. (58 + 1/3)/100 doesn't look a lot better. 7/12 seems reasonable.)
Multiplying with odds: 1:2 x 1:3 = 1:6 = 1/7.
Adding: 1:2 + 1:3 = ? 3 worlds + 4 worlds = 7, so 2:5? Double checking: 1/3 + 1/4 = (4+3)/12 = 7/12
a:b + c:d = ac+bc:bd
But do they need the same amount of computation?
The difference between the two is literally a single summation, so... yeah?
If they didn’t need exactly the same amount of information I would be very interested in what kind of math wizardry is involved.