Definition: Odds ratio, prior and posterior odds

Index: The Book of Statistical Proofs ▷ General Theorems ▷ Bayesian statistics ▷ Bayesian inference ▷ Odds ratios

Definition: Let $A_1$, $A_2$ and $B$ be arbitrary statements about random variables where $A_1$ and $A_2$ are mutually exclusive, but not nessarily collectively exhaustive.

Then, the “odds” or “prior odds ratio” in favor of $A_1$ against $A_2$ is defined as the ratio of the probabilities of $A_1$ and $A_2$, unconditional on $B$

$$\label{eq:odds-prior} \frac{p(A_1)}{p(A_2)}$$

and the “posterior odds ratio” in favor of $A_1$ against $A_2$ is defined as the ratio of the probabilities of $A_1$ and $A_2$, conditional on $B$

$$\label{eq:odds-post} \frac{p(A_1|B)}{p(A_2|B)}$$

where $p(\cdot)$ is a probability mass function or probability density function. Prior and posterior odds ratio are related to each other via the Bayes factor through Bayes’ rule.

 

Sources:

• Wikipedia (2025): "Odds ratio"; in: Wikipedia, the free encyclopedia, retrieved on 2025-02-06; URL: https://en.wikipedia.org/wiki/Odds_ratio.

Metadata: ID: D215 | shortcut: odds | author: JoramSoch | date: 2025-02-06, 16:14.