Odds

Odds ratios express relative belief: if the odds for X versus Y are 3:2, this expresses that we think that proposition X is 1.5 times as likely as proposition Y.

Like all ratios, since odds ratios express a relative quantity, multiplying all elements of the ratio by a positive scalar factor does not change the structure that the ratio expresses. $X:Y=cX:cY,c>0.$ 9:6 expresses the same ratio as 3:2.

An odds ratio can contain any number of components. In the odds ratio 3:2:6, the first component is asserted to be half as probable as the fifth component.

A ratio with only two elements can potentially be viewed as a dimensionless scalar quantity in the range $(0,+\infty )$. If the odds of Andrew Jackson becoming President are 0.75 to 0.25 in favor, we can also say that Andrew Jackson is 3 times as likely to become President as not. Or if the odds were 0.4 to 0.6, we could say that Andrew Jackson was 2/3rds as likely to become President as not.

When an odds ratio exhausts all propositions under consideration, we can convert the individual components to probabilities by normalizing the components so that they sum to 1. This in turn can be done by summing all the components of the ratio, then dividing each component by the sum:

$(x_1:x_2:\cdots :x_N)=(\frac{x_1}{{\sum }_{i=1}^N{x}_i}:\frac{x_2}{{\sum }_{i=1}^N{x}_i}:\cdots :\frac{x_N}{{\sum }_{i=1}^N{x}_i})$

E.g., to obtain a ratio of probabilities from the odds ratio 3:2, we write:

$3:2=\frac{3}{3+2}:\frac{2}{3+2}=0.6:0.4$

If we begin from an ordinary probability $P\left(X\right)$, we immediately obtain the odds for $X$ vs. $\neg X$ as $\left(P\right(X):P(\neg X\left)\right)=\left(P\right(X):(1-P\left(X\right)\left)\right).$

Odds ratios are an exceptionally convenient form to express belief updates using Bayes's Rule, since the prior odds can be term-by-term multiplied by a likelihood ratio to yield the posterior odds.