# Odds are not easier

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 14 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 17.

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 13 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 13.

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 + 13)/​100 doesn’t look a lot bet­ter. 712 seems rea­son­able.)

Mul­ti­ply­ing with odds: 1:2 x 1:3 = 1:6 = 17.

Ad­ding: 1:2 + 1:3 = ? 3 wor­lds + 4 wor­lds = 7, so 2:5? Dou­ble check­ing: 13 + 14 = (4+3)/​12 = 712

a:b + c:d = ac+bc:bd

• 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.

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.