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You are viewing revision 1.0.0, last edited by DanielLC

Odds ratios are an alternate way of expressing probabilities, which simplifies the process of updating them with new evidence. The odds ratio of A is P(A)/P(¬A).

$P(A|B) = P(B|A)\frac{P(A)}{P(B)}$

$P(\neg A|B) = P(B|\neg A)\frac{P(\neg A)}{P(B)}$

$\frac{P(A|B)}{P(\neg A|B)} = \frac{P(B|A)}{P(B|\neg A)}\frac{P(A)}{P(\neg A)}$

Thus, in order to find the posterior odds ratio $\frac{P(A|B)}{P(\neg A|B)}$, one simply multiplies the prior odds ratio $\frac{P(A)}{P(\neg A)}$ by the likelihood ratio $\frac{P(B|A)}{P(B|\neg A)}$.

Odds ratios are commonly written as the ratio of two numbers separated by a colon. For example, if P(A) = 2/3, the odds ratio would be 2, but this would most likely be written as 2:1.

The relation between odds ratio, a:b, and probability, p is as follows:

*a* : *b* = *p* : (1 − *p*) $p = \frac{a}{a+b}$

Suppose you have a box that has a 5% chance of containing a diamond. You also have a diamond detector that beeps two thirds of the time if there is a diamond, and one third of the time if there is not. You wave the diamond detector over the box and it beeps.

The prior odds of the box containing a diamond are 1:19. The likelihood ratio of a beep is 2/3:1/3 = 2:1. The posterior odds are 1:19 * 2:1 = 2:19. This corresponds to about a probability of 2/21, which is about 0.095 or 9.5%.